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CONTENTS PREFACE TO "MEANS AND THEIR INEQUALITIES" xiii PREFACE TO THE HANDBOOK xv BASIC REFERENCES xvii NOTATIONS xix 1. Referencing xix 2. Bibliographic References xix 3. Symbols for Some Important Inequalities xix 4. Numbers, Sets and Set Functions xx 5. Intervals xx 6. n-tuples xxi 7. Matrices xxii 8. Functions xxii 9. Various xxiii A LIST OF SYMBOLS xxiv AN INTRODUCTORY SURVEY xxvi CHAPTER I INTRODUCTION 1 1. Properties of Polynomials 1 1.1 Some Basic Results 1 1.2 Some Special Polynomials 3 2. Elementary Inequalities 4 2.1 Bernoulli's Inequality 4 2.2 Inequalities Involving Some Elementary Functions 6 3. Properties of Sequences 11 3.1 Convexity and Bounded Variation of Sequences 11 3.2 Log-convexity of Sequences 16 3.3 An Order Relation for Sequences 21 4. Convex Functions 25 4.1 Convex Functions of Single Variable 25 4.2 Jensen's Inequality 30 4.3 The Jensen-Steffensen Inequality 37 4.4 Reverse and Converse Jensen Inequalities 43 4.5 Other Forms of Convexity 48 4.5.1 Mid-point Convexity 48 4.5.2 Log- convexity 48 4.5.3 A Function Convex with respect to Another Function 49 vii

4.6 Convex Functions of Several Variables 50 4.7 Higher Order Convexity 54 4.8 Schur Convexity 57 4.9 Matrix Convexity 58 CHAPTER II THE ARITHMETIC, GEOMETRIC AND HARMONIC MEANS 60 1. Definitions and Simple Properties 60 1.1 The Arithmetic Mean 60 1.2 The Geometric and Harmonic Means 64 1.3 Some Interpretations and Applications 66 1.3.1 A Geometric Interpretation 66 1.3.2 Arithmetic and Harmonic Means in Terms of Errors 67 1.3.3 Averages in Statistics and Probability 67 1.3.4 Averages in Statics and Dynamics 68 1.3.5 Extracting Square Roots 68 1.3.6 Cesaro Means 69 1.3.7 Means in Fair Voting 70 1.3.8 Method of Least Squares 70 1.3.9 The Zeros of a Complex Polynomial 70 2. The Geometric Mean-Arithmetic Mean Inequality 71 2.1 Statement of the Theorem 71 2.2 Some Preliminary Results 72 2.2.1 (GA) with η = 2 and Equal Weights 72 2.2.2 (GA) with η = 2, the General Case 76 2.2.3 The Equal Weight Case Suffices 80 2.2.4 Cauchy's Backward Induction 81 2.3 Some Geometrical Interpretations 82 2.4 Proofs of the Geometric Mean-Arithmetic Mean Inequality 84 2.4.1 Proofs Published Prior to 1901. Proofs (i)-(vii) 85 2.4.2 Proofs Published Between 1901 and 1934. Proofs (viii)-(xvi) .. 89 2.4.3 Proofs Published Between 1935 and 1965. Proofs (xvii)-(xxxi) .. .92 2.4.4 Proofs Published Between 1966 and 1970. Proofs (xxxii)-(xxxvii) 101 2.4.5 Proofs Published Between 1971 and 1988. Proofs (xxxviii)-(lxii) 104 2.4.6 Proofs Published After 1988. Proofs (lxiii)-(lxxiv) 114 2.4.7 Proofs Published In Journals Not Available to the Author 118 2.5 Applications of the Geometric Mean-Arithmetic Mean Inequality 119 2.5.1 Calculus Problems 119 2.5.2 Population Mathematics 120 2.5.3 Proving other Inequalities 120 2.5.4 Probabilistic Applications 124 3. Refinements of the Geometric Mean-Arithmetic Mean Inequality ... 125 3.1 The Inequalities of Rado and Popoviciu 125 3.2 Extensions of the Inequalities of Rado and Popoviciu 129 3.2.1 Means with Different Weights 129 3.2.2 Index Set Extensions 132 viii

3.3 A Limit Theorem of Everitt 133 3.4 Nanjundiah's Inequalities 136 3.5 Kober-Diananda Inequalities 141 3.6 Redheffer's Recurrent Inequalities 145 3.7 The Geometric Mean-Arithmetic Mean Inequality with General Weights 148 3.8 Other Refinements of Geometric Mean-Arithmetic Mean Inequality 149 4. Converse Inequalities 154 4.1 Bounds for the Differences of the Means 154 4.2 Bounds for the Ratios of the Means 157 5. Some Miscellaneous Results 160 5.1 An Inductive Definition of the Arithmetic Mean 160 5.2 An Invariance Property 160 5.3 Cebisev's Inequality 161 5.4 A Result of Diananda 165 5.5 Intercalated Means 166 5.6 Zeros of a Polynomial and Its Derivative 170 5.7 Sanson's Inequality 170 5.8 The Pseudo Arithmetic Means and Pseudo Geometric Means ... 171 5.9 An Inequality Due to Mercer 174 CHAPTER III THE POWER MEANS 175 1. Definitions and Simple Properties 175 2. Sums of Powers 178 2.1 Holder's Inequality 178 2.2 Cauchy's Inequality 183 2.3 Power sums 185 2.4 Minkowski's Inequality 189 2.5 Refinements of the Holder, Cauchy and Minkowski Inequalities . 192 2.5.1 A Rado type Refinement 192 2.5.2 Index Set Extensions 193 2.5.3 An Extension of Kober-Diananda Type 194 2.5.4 A Continuum of Extensions 194 2.5.5 Beckenbach's Inequalities 196 2.5.6 Ostrowski's Inequality 198 2.5.7 Aczel-Lorentz Inequalities 198 2.5.8 Various Results 199 3. Inequalities Between the Power Means 202 3.1 The Power Mean Inequality 202 3.1.1 The Basic Result 203 3.1.2 Holder's Inequality Again 211 3.1.3 Minkowski's Inequality Again 213 3.1.4 Cebisev's Inequality 215 3.2 Refinements of the Power Mean Inequality 216 3.2.1 The Power Mean Inequality with General Weights 216 3.2.2 Different Weight Extension 216 3.2.3 Extensions of the Rado-Popoviciu Type 217 ix

3.2.4 Index Set Extensions 220 3.2.5 The Limit Theorem of Everitt 225 3.2.6 Nanjundiah's Inequalities 225 4. Converse Inequalities 229 4.1 Ratios of Power Means 230 4.2 Differences of Power Means 238 4.3 Converses of the Cauchy, Holder and Minkowski Inequalities 240 5. Other Means Defined Using Powers 245 5.1 Counter-Harmonic Means 245 5.2 Generalizations of the Counter-Harmonic Means 248 5.2.1 Gini Means 248 5.2.2 Bonferroni Means 251 5.2.3 Generalized Power Means 251 5.3 Mixed Means 253 6. Some Other Results 256 6.1 Means on the Move 256 6.2 Hlawka-type inequalities 258 6.3 p-Mean Convexity 260 6.4 Various Results 260 CHAPTER IV QUASI-ARITHMETIC MEANS 266 1. Definitions and Basic Properties 266 1.1 The Definition and Examples 266 1.2 Equivalent Quasi-arithmetic Means 271 2. Comparable Means and Functions 273 3. Results of Rado-Popoviciu Type 280 3.1 Some General Inequalities 280 3.2 Some Applications of the General Inequalities 282 4 Further Inequalities 285 4.1. Cakalov's Inequality 286 4.2 A Theorem of Godunova 288 4.3 A Problem of Oppenheim 290 4.4 Ky Fan's Inequality 294 4.5 Means on the Move 298 5. Generalizations of the Holder and Minkowski Inequalities 299 6. Converse Inequalities 307 7. Generalizations of the Quasi-arithmetic Means 310 7.1 A Mean of Bajraktarevic 310 7.2 Further Results 316 7.2.1 Deviation Means 316 7.2.2 Essential Inequalities 317 7.2.3 Conjugate Means 320 7.2.4 Sensitivity of Means 320 CHAPTER V SYMMETRIC POLYNOMIAL MEANS 321 1. Elementary Symmetric Polynomials and Their Means 321 2. The Fundamental Inequalities 324 3. Extensions of S(r;s) of Rado-Popoviciu Type 334 χ

4. The Inequalities of Marcus & Lopes 338 5. Complete Symmetric Polynomial Means; Whiteley Means 341 5.1 The Complete Symmetric Polynomial Means 341 5.2 The Whiteley Means 343 5.3 Some Forms of Whiteley 349 5.4 Elementary Symmetric Polynomial Means as Mixed Means 356 6. The Muirhead Means 357 7. Further Generalizations 364 7.1 The Hamy Means 364 7.2 The Hayashi Means 365 7.3 The Biplanar Means 366 7.4 The Hypergeometric Mean 366 CHAPTER VI OTHER TOPICS 368 1. Integral Means and Their Inequalities 368 1.1 Generalities 368 1.2. Basic Theorems 370 1.2.1 Jensen, Holder, Cauchy and Minkowski Inequalities 370 1.2.2 Mean Inequalities 373 1.3 Further Results 377 1.3.1 A General Result 377 1.3.2 Beckenbach's Inequality; Beckenbach-Lorentz Inequality .... 378 1.3.3 Converse Inequalities 380 1.3.4 RyfFs Inequality 380 1.3.5 Best Possible Inequalities 381 1.3.6 Other Results 382 2. Two Variable Means 384 2.1 The Generalized Logarithmic and Extended Means 385 2.1.1 The Generalized Logarithmic Means 385 2.1.2 Weighted Logarithmic Means of n-tuples 391 2.1.3 The Extended Means 393 2.1.4 Heronian, Centroidal and Neo-Pythagorean Means 399 2.1.5 Some Means of Haruki and Rassias 401 2.2 Mean Value Means 403 2.2.1 Lagrangian Means 403 2.2.2 Cauchy Means 405 2.3 Means and Graphs 406 2.3.1 Alignment Chart Means 406 2.3.2 Functionally Related Means 407 2.4 Taylor Remainder Means 409 2.5 Decomposition of Means 412 3. Compounding of Means 413 3.1 Compound means 413 3.2 The Arithmetico-geometric Mean and Variants 417 3.2.1 The Gaussian Iteration 417 3.2.2 Other Iterations 419 4. Some General Approaches to Means 420 xi

4.1 Level Surface Means 420 4.2 Corresponding Means 422 4.3 A Mean of Galvani 423 4.4 Admissible Means of Bauer 423 4.5 Segre Functions 425 4.6 Entropic Means 427 5. Mean Inequalities for Matrices 429 6. Axiomatization of Means 435 BIBLIOGRAPHY 439 Books 439 Papers 444 NAME INDEX 511 INDEX 525 xii

'Means and Their Inequalities"-Preface There seems to be two types of books on inequalities. On the one hand there are treatises that attempt to cover all or most aspects of the subject, and where an attempt is made to give all results in their best possible form, together with either a full proof or a sketch of the proof together with references to where a full proof can be found. Such books, aimed at the professional pure and applied mathematician, are rare. The first such, that brought some order to this untidy field, is the classical "Inequalities" of Hardy, Littlewood L· Polya, published in 1934. Important as this outstanding work was and still is, it made no attempt at completeness; rather it consisted of the total knowledge of three front rank mathematicians in a field in which each had made fundamental contributions. Extensive as this combined knowledge was there were inevitably certain lacunae; some important results, such as Steffensen's inequality, were not mentioned at all; the works of certain schools of mathematicians were omitted, and many important ideas were not developed, appearing as exercises at the ends of chapters. The later book "Inequalities" by Beckenbach L· Bellman, published in 1961, repairs many of these omissions. However this last book is far from a complete coverage of the field, either in depth or scope. A much more definitive work is the recent "Analytic Inequalities" by Mitrinovic, (with the assistance of Vasic), published in 1970, a work that is surprisingly complete considering the vast field covered. On the other hand there are many works aimed at students, or non-mathematicians. These introduce the reader to some particular section of the subject, giving a feel for inequalities and enabling the student to progress to the more advanced and detailed books mentioned above. Whereas the advanced books seem to exist only in English, there are excellent elementary books in several languages: "Analytic Inequalities" by Kazarinoff, "Geometric Inequalities" by Bottema, Djordjevic, Janic &; Mitrinovic in English; "Nejednakosti" by Mitrinovic, "Sredine" by Mitrinovic &: Vasic in Serbo-croatian, to mention just a few. Included in this group although slightly different are some books that list all the inequalities of a certain type—a sort of table of inequalities for reference. Several books by Mitrinovic are of this type. Due to the breadth of the field of inequalities, and the variety of applications, none of the above mentioned books are complete on all of the topics that they take up. Most inequalities depend on several parameters, and what is the most natural domain for these parameters is not necessarily obvious, and usually it is not the widest possible range in which the inequality holds. Thus the author, even xin

the most meticulous, is forced to choose; and what is omitted from the conditions of an inequality is often just what is needed for a particular application. What appears to be needed are works that pick some fairly restricted area from the vast subject of inequalities and treat it in depth. Such coherent parts of this discipline exist. As Hardy, Littlewood & Polya showed, the subject of inequalities is not just a collection of results. However, no one seems to have written a treatise on some such limited but coherent area. The situation is different in the collection of elementary books; several deal with certain fairly closely defined areas, such as geometric inequalities, number theoretic inequalities, means. It is the last mentioned area of means that is the topic of this book. Means are basic to the whole subject of inequalities, and to many applications of inequalities to other fields. To take one example: the basic geometric mean-arithmetic mean inequality can be found lurking, often in an almost impenetrable disguise, behind inequalities in every area. The idea of a mean is used extensively in probability and statistics, in the summability of series and integrals, to mention just a few of the many applications of the subject. The object of this book is to provide as complete an account of the properties of means that occur in the theory of inequalities as is within the authors' competences. The origin of this work is to be found in the much more elementary "Sredine" mentioned above, which gives an elementary account of this topic. A full discussion will be given of the various means that occur in the current literature of inequalities, together with a history of the origin of the various inequalities connecting these means1. A complete catalogue of all important proofs of the basic results will be given as these indicate the many possible interpretations and applications that can be made. Also, all known inequalities involving means will be discussed. As is the nature of things, some omissions and errors will be made: it is hoped that any reader who notices any such will let the authors know, so that later editions can be more complete and accurate. An earlier version of this book was published in 1977 in Serbo-croatian under the title " Sredine i sa Njima Povezane Nejednakosti". The present work is a complete revision, and updating of that work. The authors thank Dr J. E. Pecaric of the University of Belgrade Faculty of Civil Engineering for his many suggestions and contributions. Vancouver & Belgrade 1988 Although not mentioned in this preface the book was devoted to discrete mean inequalities and did not discuss in any detail integral mean inequalities, matrix mean inequalities or mean inequalities in general abstract spaces. This bias will be followed in this book except in Chapter VI. XIV

PREFACE TO THE HANDBOOK Since the appearance of Means and Their Inequalities the deaths of two of the authors have occurred. The field of inequalities owes a great debt to Professor Mitrinovic and his collaborator for many years, Professor Vasic. Over a lifetime Professor Mitrinovic devoted himself to inequalities and to the promotion of the field. His journal, Univerzitet и Beogradu Publikacije Elektrotehnickog Fakulteta. Serija Matematika i Fizika, the "i Fizika' was dropped in the more recent issues, has in all of its volumes, from the first in the early fifties, devoted most of its space to inequalities. In addition his enthusiasm has resulted in a flowering of the study both by his students, P. M. Vasic, J. E. Pecaric to mention the most notable, and by many others. The uncertain situation in the former Yugoslavia has lead to many of the researchers situated in that country moving to institutions all over the world. There are now more journals devoted to inequalities, such as the Journal for Inequalities and Applications and Mathematical Inequalities and Applications, as well as many that devote a considerable portion of their pages to inequalities, such as the Journal of Mathematical Analysis and Applications; in addition mention must be made of the electronic Journal of Inequalities in Pure and Applied Mathematics based on the website http://rgmia.vu.edu.au and under the editorship of S. S. Dragomir. This website has in addition many monographs devoted to inequalities as well as a data base of inequalities, and mathematicians working in the field. Another welcome change has been the many contributions from Asian mathematicians. While there have always been results from Japan, in recent years there has been a considerable amount of work from China, Singapore, Malaysia and elsewhere in that region. It was taken for granted in the earlier Preface that anyone reading this book would not only be interested in inequalities but would be aware of their many applications. However it would not be out of place to emphasize this by quoting from a recent paper; [Guo L· Qi). "It is well known that the concepts of means and their inequalities not only are basic and important concepts in mathematics, (for example, some definitions of norms are often special means2), and have explicit geometric More precisely "... certain means are related to norms and metrics." See III 2 4, 2 5.7 VI 2.2.1 XV

meanings3, but also have applications in electrostatics4, heat conduction and chemistry5. Moreover, some applications to medicine6 have been given recently." Due to the extensive nature of the revision in the second edition and the large amount of new material it has seemed advisable to alter the title but this handbook could not have been prepared except for the basic work done by my late colleagues and I only hope that it will meet the high standards that they set. In addition I want to thank my wife Georgina Bullen who has carefully proofread the non-mathematical parts of the manuscript, has suffered from computer deprivation while I monopolized the screen, and without whose support and help the book would have appeared much later and in a poorer form. P. S. Bullen Department of Mathematics University of British Columbia Vancouver ВС Canada V6T 1Z2 bullen@math.ubc.ca 3 See II 1.1, 1.3.1; [Qi & Luo]. 4 See[Polya & Szego 1951). See[Walker, Lewis & McAdams], [Tettamanti, Sarkany, Kralik &:Stomfai; Tettamanti L· Stomfai] 6 See [Ding]. XVI

BASIC REFERENCES There are some books on inequalities to which frequent reference will be made and which will be given short designations. [AI] Mitrinovic, D. S., with Vasic P. M. Analytic Inequalities, Springer-Verlag, Berlin, 1970. [BB] Beckenbach, E. F. & Bellman, R. Inequalities, Springer-Verlag, Berlin, 1961. [HLP] Hardy, G. H., Littlewood, J. E. & Polya, G. Inequalities, Cambridge University Press, Cambridge, 1934. [MI] Bullen, P.S., Mitrinovic, D. S L· Vasic P. M. Means and Their Inequalities, D.Reidel. Dordrecht, 1988. [The first edition of this handbook.] [MO] Marshall, A. W. & Olkin, I. Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979. Many inequalities can be placed in a more general setting. We do not follow that direction in this book but find the following an invaluable reference. Much of the material is readily translated to our simpler less abstract setting. [PPT] Pecaric, J. E., Proschan, F. & Tong, Y. L. Convex Functions, Partial Orderings and Statistical Applications, Academic Press Inc., 1992. There are two books that are referred to frequently in certain parts of the book and for which we also introduce short designations. [B2] Borwein, J. M. & Borwein, P. B. Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity, John Wiley and Sons, New York, 1987. [RV] Roberts, A. W. L· Varberg, D. E. Convex Functions, Academic Press, New York-London, 1973. In addition there are the two following references. The first is a ready source of information on any inequality, and the second is in a sense a continuation of [AI] and [BB] above, being a report on recent developments in various areas of inequalities. xvn

[DI] Bullen, P. S. A Dictionary of Inequalities, Addison-Wesley Longman, London, 1998.7 [MPF] Mitrinovic, D. S., Pecaric, J. E. L· Fink, A. M. Classical and New Inequalities in Analysis, D Reidel, Dordrecht, 1993. There are many other books on inequalities and many books that contain important and useful sections on inequalities. These are listed in Bibliography Books. From time to time conferences devoted to inequalities have published their proceedings. In particular, there are the proceedings of three symposia held in the United States, and of seven international conferences held at Oberwolfach. 11(1965), 12 (1967), 13 (1969) Inequalities, Inequalities II, Inequalities III, Proceedings of the First, Second and Third Symposia on Inequalities, 1965, 1967, 1969; Shisha, O., editor, Academic Press, New York, 1967, 1970, 1972. GI1(1976), GI2 (1978), GI3 (1981), GI4 (1984), GI5 (1986), GI6 (1990), GI7 (1995) General Inequalities Volumes 1-7, Proceedings of the First-Seventh International Conferences on General Inequalities, Oberwolfach, 1976, 1978, 1981, 1984, 1986, 1990, 1995; Beckenbach, E. F., Walter, W., Bandle, C., Everitt, W. N., Losonczi, L., [Eds.], International Series of Numerical Mathematics, 41, 47, 64, Birkhauser Verlag, Basel, 1978, 1980, 1983, 1986,1987, 1992, 1997. Individual papers in these proceedings, referred to in the text, are listed under the various authors with above shortened references. Finally there are two general references. [EMI], [EM2], [EM3], [EM4], [EM5], [EM7], [EM8], [EM9],[EM10], [EMSuppI], [EMSuppII], [EMSuppIII]; Hazelwinkel, M., [Ed.] Encyclopedia of Mathematics, vol.1-10, suppl. I-III, Kluwer Academic Press, Dordrecht, 1988-2001. [CE] Weisstein, E. W. CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, Boca Raton, 1998. Additions and corrections can be found at http://rgmia.vu.edu.au/monographs/bullen.html. XV111

NOTATIONS 1 Referencing Theorems, definitions, lemmas, corollaries are numbered consecutively in each section; the same is true of formulae. Remarks and Examples are numbered, using Roman numerals, consecutively in each subsection, and in each sub-subsection. In the same chapter references list the section, (subsection, sub-section, in the case of remarks and examples), followed by the detail: thus 3 Theorem 2, 4(6), or 1.2 Remark (6). Footnotes are numbered consecutively in each chapter and so are referred to by number in that chapter: thus 1.3.1 Footnote 1. References to other chapters are as above but just add the chapter number; thus I 3 Theorem 2(a), II 4(6), IV 1.2 Remark (6), I 1.3.1 Footnote 1. Although there are references for all names and all bibliography entries, in the case of a name occurring frequently, for instance Cauchy, only the most important instances will be mentioned; further names in titles of the basic references are usually not referenced, thus Hardy in [HLP]. 2 Bibliographic References Some have been given a shortened form; see Basic References. Others are standard, the name, followed by a year if there is ambiguity, or the year with an additional letter, such as 1978a, if there is more than one any given year. Joint authorship is referred to by using &, thus Mitrinovic L· Vasic. 3 Symbols for Some Important Inequalities Certain inequalities are referred to by a symbol as they occur frequently. (B) Bernoulli's inequality I 2.1 Theorem 1 (C) Cauchy's inequality III 2.2 (GA) Geometric-Arithmetic Mean inequality II 2.1 Theorem 1 (H) Holder's inequality III 2.1 Theorem 1 (HA) Harmonic- Arithmetic Mean Inequality II 2.1 Remark(i) (J) Jensen 's inequality 14.2 Theorem 12 (M) Minkowski's inequality III 2.4 Theorem 9 (P) Popviciu's inequality II 3.1 Theorem 1 (R) Rado's inequality II 3.1 Theorem 1 (r;s) Power Mean inequality III 3.1.1 Theorem 1 S(r;s)... .Elementary Symmetric Polynomial Mean inequality V 2 Theorem 3(b) xix

(Τ), (ΎΝ) Triangle inequality III 2.4 . Integral analogues of these means, when they exist, will be written (J)-/, etc; see VI 1.2.1, 1.2.2.; and (~B), etc. will denote the opposite inequalities; see 9 below. 4 Numbers, Sets and Set Functions Z, Q,R, С are standard notations for the sets of integers, rational numbers, real numbers, and complex numbers respectively. The set of extended real numbers, R U {±oo}, is written R. Less standard are the following: Ν = {η; η e Z, andn > 0} = {0,1,2,...}, N* = N \ {0} = {1, 2,...}, R* =R\{0}, R+ = {x;x eRandrr > 0}, R^ = R+ \ {0} = {x;x eRandz > 0}, Q* = Q\{0}, Q+ = {x;xEQandx>0}, Ql = Q+\{0} = {x]xeQtmdx> 0}. The non-empty subsets of N, or N*, are called index sets, the collection of these is written X. If ρ e R*, ρ 7^ 1, the conjugate index ofp, written ρ', is defined by ρ 11 ρ' = -, equivalently —I—- = 1. ρ - 1 Ρ Ρ Note that (p'Y = p> ρ > ι -£=> ρ' > ι, ρ > о and ρ' > ο ^^ ρ > ι, or ρ7 > ι. A real function φ defined on sets, a set function, is said to be super-additive, respectively log-super-additive, if for any two non-intersecting sets, I, J say, in its domain φ(Ι U J) > φ(Ι) + φ{3), respectively, φ(Ι U J) > φ{Ι)φ{3). If these inequalities are reversed the function is said to be sub-additive, respectively log-sub-additive. A set function / is said to be increasing if J С J =>> /CO < /(«0> and if the reverse inequality holds it is said to be decreasing. The image of a set A by a function f will be written f[A}; that is f[A] = {у; у = /CO, χ e A}. A set Ε is called a neighbourhood of a point χ if for some open interval ]o, b[ we have ж Ε [α, 6[C J5. 5 Intervals Intervals in R are written [a, b], closed; ]a,b[, open etc. In addition we have the unbounded intervals [a, oo[, ] — oo, a], etc; and of course ] — oo, oo[= R, [—oo, oo] = R, [0,oo[= R_|_, ]0, oo[= RUj_. The symbol (a, b) is reserved for 2-tuples, see the о next paragraph. If i" is any interval then i" denotes the open interval with the same end-points as i". xx

6 η-tuples If di e R, C, 1 < i < n, then we write a for the, ordered, real, complex, n-tuple (αϊ,..., αη) with these elements or entries; so if α is a real η-tuple α Ε Rn. The usual vector notation is followed. In addition the following conventions are used. (i) For suitable functions /, g etc, /(α) = (/(αχ),..., /(an)), and g(a,b) = (g(a\, bi),... ,#(αη, &n)) etc. This useful convention conflicts with the standard notation for functions of several variables, /(a) = /(ai,..., an), but the context will make clear which is being used. So: a2 = (af,...,a^), ab = (aibi,..., anbn); but max a = maxi<Kna,, and min a = mini<i<n &%· (ii) a < b means that ai < bi, 1 < г < η. In addition m < a< Μ means that m < di < M, 1 < г < n. The extensions to the other inequality signs is obvious. In particular if a > 0, a > 0 etc, we say that the η-tuple is positive, non-negative etc. The set of non-negative η-tuples is written R™, and the set of positive n-tuples (м;)п (iii) An η-tuple all of whose elements are all zero except the г-th that is equal to 1 is written e^. The η-tuples e^, 1 < г < n, form the standard basis of Rn. As usual if η = 2,3 these basis vectors are written ег = г, е2 = j, е% = к. (iv) e is the η-tuple each of whose entries is equal to 1 ; and 0, or just 0, is the η-tuple each of whose entries is equal to 0. (v) If 1 < г < η then o!i denotes the (n — l)-tuple obtained from α by omitting the element a^. (vi) a ~ b, α is proportional to 6, means that for some λ, μ G R, not both zero, λα + /ib = 0; that is the two η-tuples are linearly dependent. (vii) If α^ = c, 1 < г < η, we say that a is constant, or is a constant. (viii) For Akan and see I 3.1. An η-tuple is called an arithmetic progression if Δ1α^, 1 < к < η — 1, is constant, equivalently if Δ2α& = 0,1 < к < η — 2, (ix) For а*Ь see I 3.2 Definition 4. (x) For a-<b see I 3.3 Definition 11. (xi) Given an η-tuple w. we will write Wk = Σ^=1^, 1 < г < η; and if necessary Wo = 0· More generally if υ; is a sequence and / G I, an index set, see 4, we write Wj = Y,ieI Щ- (xii) If / is a function of к variables, a and n-tuple, 1 < к < η, then Σ&·./*(αΰ' · · · 5 aik) means that the summation is taken over all permutations of к elements from the αϊ,... ,αη. In the case that к = η this will be written as Σ!/(«)· (xiii) The inner product of two η-tuples, a,b, written (а,Ь), is Σ™=1α$ι if the η-tuples are real and Σ™=1 оцЬг if the η-tuples are complex. In both cases Where appropriate the above notations will be used for sequences α = (αϊ, аг,...), provided the relevant series converge. xxi

7 Matrices An πι χ η matrix is A = (a^·) i<i<m; the a^· are the entries and if they are real, l<j<n complex then A is said to be real, complex. The transpose of A is AT = (ац) i<i<m ; l<j<n A* = (aji)i<z<m is the conjugate transpose of A. l<j<n If A = AT then A is symmetric; if A = A* then A is Hermitian-, in these cases of course m = η and the matrix is said to be square. The family of all η χ η Hermitian matrices will be written Hn', the subset of positive definite η xn Hermitian matrices is written 7Y+. If А, В e Hn we write A < В if В —A is positive semi-definite, and A < В if В — A is positive definite. This defines an order on Hn called the Loewner ordering. A square matrix D with all non-diagonal elements zero is called a diagonal matrix; if the diagonal elements are Q"ii = di, 1 < г < η, we write D = D(d) = D(du ..., α^). li d = e the diagonal matrix is called a unit matrix, usually written i", or In if it is important to show that it is an η χ η unit matrix8. The square matrix all of whose entries are equal to 1 is written J, or if the order needs to be noted Jn. A matrix of any order all of whose entries are zero, a zero matrix, will be written O, the order being understood from the context. If A E Hn with eigenvalues λ, then all of its eigenvalues real and A — U*D(X)U where U is a unitary matrix, that is UU* = I; so for a unitary matrix Ό~ι = U*. If A E Hn and / is a real-values function defined on an interval that contains the eigenvalues of A we define the associated matrix function of order n, using the same symbol, / : Hn i-> Hn by f{A) = UD(f(\))U*; see [MO p.462]. Further (f(A)a,a) = ZL· \bi\2f(*i) where b = Ua. 8 Functions A function of η variables / is said to be symmetric if its domain is symmetric and if its value is not altered by a permutation of the variables. Precisely, if / : D >—► R and (i) D С Шп, (ii) (al5..., αη) Ε D => (α^,..., α^η) Ε £) for every permutation (г\,. .. ,гп) of (1,... ,n), (Hi) /(α^,... ,ain) = /(αϊ,... ,αη) for every permutation (ii,..., in) of (1,. .., n), then / is said to be symmetric. A related concept is that of an almost symmetric function: this is a function of η variables and η parameters, defined on a symmetric domain, that is invariant under the simultaneous permutation of the variables and the parameters. Examples(i) The arithmetic mean with equal weights, /(α) = (αϊ Η han)/n, is symmetric; the weighted arithmetic mean, /(a) = /(a;w;) = (w\a\ + · · · 4- WnO"n)/(wi Η 4- гип), is almost symmetric; see II 1.1. The Gamma or factorial function is denoted by ж!; that is ж! = Г(ж4-1). The identity function is written *,, and then the power functions ls,s eR*. That is l{x) = x; ls(x) = xs; This conflicts with the use of I to denote an interval but in practice there will be no confusion. ХХ11

the domain being clear from the context. The maximum function is defined as usual by max{/, g]{x) = тах{/(ж),^(ж)}: also, ж+ = тах{ж, 0}; x~ = max{—x, 0}; χ = x+ — x~, |x| = x+ + x~. By analogy if / is any real-valued function then /+ = max{/, 0} = max of; when / = /+-/-, 1/1 = /+ + /-. The function that is equal to 1 on the set A and to zero off A, the indicator function9 of A is written 1a- That is The integral or integer part function [■] : R t—> Ζ is defined by: [ж] = η, where η Ε Ζ is the unique integer such that η < χ < η + 1. 9 Various Proofs begin with a square, D, on the left, and end with one on the right. If (n) denotes an inequality then (~n) will denote the inequality obtained by changing the inequality sign in (n); the opposite, inverse or reverse inequality. A different concept is the converse or complementary inequality. Such inequalities arise as follows: in general an inequality is of the form F > G or equivalently F — G > 0. Usually both F, G are continuous functions and the inequality holds on some non-compact set. If now we restrict the domain to a compact set we will get that F — G attains a maximum on that set, F — G < Λ say. Another form arises using the equivalent F/G > 1, giving on the compact set F/G < λ say. The inequalities F — G < Λ, F/G < λ are converse inequalities for the original inequality F > G. If limx_,Xo f(x)/g(x) = 0 we say that / is little-oh of g as χ —ν xq , written f(x) — о(^(ж)), χ —ν xq. In particular f{x) = o(l), χ —► xq means that limx^Xo f{x) = 0. If there is some positive constant Μ such that for all χ in some neighbourhood of xq \f(x)\ < Mg(x) we say that / is big-oh of д for χ as χ —ν xq , written f{x) = 0[д(х)) χ -^ xq. In particular f(x) = O(l), x —► жо means that / is bounded at xq. If f{x) = 0{g{x)) and g{x) = 0[f{x)) for x in a given set then we say that f(x) and g(x) are of the same order of magnitude for χ in the given set, written f{x) ~ g(x), χ —► xq. Means are denoted throughout by Gothic letters: 21, <3,$)... etc. Classes of functions are denoted throughout by calligraphic letters; thus С (I), the space of functions continuous on /, Cn(7), the space of functions with continuous nth order derivatives on /, etc. This function is also called the characteristic function of A. XX111

A LIST OF SYMBOLS The following is a list of the symbols used in the text. N,N* xx R*,R+,]R; xx q*,q+,q; xx X xx p' xx f[A] xx [a, b], ]a, b[, etc xx [a, oo[, ] — oo, a] xx о I xx /(a),p(a,b) xxi max a, min a xxi a<b,m<a<M xxi Щ, (R+)n xxi e^hjjk xxi e , 0 xxi a^ xxi a~b xxi Δαη, Akan, Δαη, Akan xxi a-kb xxi b -< a xxi Efc!/K>--->a*fc) ™ ЕШ ™ Wn,Wj xxi (a, b) xxi fe /], [a; /]n, [a0,. .., an; /] 54 a ^. 68 a,Wm 132 |a| 357 Un.Ui xxii D(d) = D(du ... ,dn) xxii I,In,J,Jn,0 xxii xl xxii l(x), is (x) xxii ж+, x~, /+ xxiii 1a (x) xxiii [x] xxiii f+(Uo',v) 51 Ш 350 /* 381 f <g 381 D xxiii (~ ·) xxiii Яп(а), 2U(a; ш), ^fe ш) 60, 63 Sln(a*,l < г < n),2tn(ai,...,an) .61 «η-ι(α) 61 21/(a; w) 132 &m(a;w) 132 Щ&ш)Ж 136 ^пЧ^ш),^"1 136 <(α), β;(a) etc 295 *n,a(a) 357 *[a,4 (/;/*) 374 2t(A,5;i),Sl(A,5) 429 2im(Ab...,Am;^) 432 ®M(a) 424 C(I),Cn(I) xxiii €n(a;w) 270 4Γ](α) 341 ckfc;^(g;a) 350 B£5(a;w) 229 £>(/) 316 XXIV

£η(α;£) 316 екг]Ы 321 ekfc;^(g;a) 350 £r's(a,b) 393 ё*\а,щ) 428 ®п(а;ш), &п(а), #(а;ш) 65 07(а;ш) 132 Фт(а\щ) 132 0(a;w),<6 136 extern), <δ_1 136 <6%q(a;w) 249 0[a,b](/;M) 374 б(ЛБ;^),б(ДВ) 429 <6m(A1,...,Arn;w) 432 Hn,Ht xxii Йп(а;ж),Йп(й),Й(а;ш) 65 Ъ[:](а]Ш) 245 (Ы)1\а) 364 %,.ь]СМ 374 Йе(а,&) 399 fi^iM) 400 S)(A,B;t),f)(A,B) 429 S)m(Au...,Am]w) 432 3{a,b) 167 3η(α;μ) 392 £(а,Ь) 167 £μ(α,6),£(α,6) 369 ££(а,Ь),£~(а,Ь) 369,370 £[р](а,Ь) 385 £п(й),£п(а;ш) 391, 392 £n(a;w) 391 Ш1И(а;ш),Я«кг1(а), ^5 Ш^г](а;ш),9ЯИ(М) 175 mn(s,t;k;a) 254 Wln(a]w) 266 Щ,п{а',ш) 269 Ш?п'(а;^),Ш?п(а;;ша;) 310, 311 mn(а; а) 357 яИ[а,ч(/;/х) 373 ^[в!ь](/;м) 374 Ш1[£М](а,Ь) 403 Ш1[^(а,Ь) 406 Ж <g> 9t(a, b),Wl®a 9t(a, b) 414 ®?=1Ш1Й*](а;ш) 416 9t£](a;w) 226 04fc](a) 323 0(р(ж)),о(р(ж)), ж->ж0 xxin Й(а,Ь) 169 Un(a]w) 175 0£в(а;ш) 229 Qkrl(a),qkr](a) 341 Qm](/;/x) 374 i2n+i(/;a;x) 7 7г(г, а), 7гп (г, а), 7г(г) 185, 186 7Z(r,a',w) 187 *H[/'n](a,b) 409 Фп\а,Ъ) 410 S(r,a),Sn(r,a),S(r) 185, 186 <S(r,a;w) 187 6n(a;w) 270 Sm{q.;w) 279 &:\a),sl\a) 322 Tn+i(/;a;x) 7 4M(a) 344 t[n^{a) 350 4Μ(α),»1Μ(α) 344 The Gothic alphabet: a, 21, b, 05, c, £, D, m,OT,n,ai,o,D,p,q2,q,n,r,^,5,6,t, T,u,il,d,QJ,m,2n,?,X,t),2},3,3. The Greek alphabet: a, /?, 7, Γ, £, Δ, e, ε, С, 77,0, θ, 6, /с, λ, Λ, μ, ζ/, ξ, Ξ, ω, π, Π, ш, ρ, ρ, σ, Σ, τ, ι?, Τ, φ, Φ, ν?, χ, ^, Ψ.ω, Ω XXV

AN INTRODUCTORY SURVEY This book is aimed at a wide audience. For those in the field of inequalities the table of contents, and the abstracts at the beginnings of chapters should suffice as a guide to the material. However for others it may be useful to indicate the basic material so as to allow the avoidance of the specialised sections. The basic material can be found in Chapters I, II, III and VI while the remaining chapters deal mainly with material that is more technical. However even in the basic chapters material of less interest to the general reader occur. Such avoidable sections are: I 3.1, 3.2, 4.5.3, 4.6, 4.8, 4.9; II 3.2-3.6, 3.8, 4, 5.4, 5.6-5.9; III 2.3, 2.5, 3.1.2-3.1.4, 3.2, 4, 5.2.3, 5.4,, 6.2-6.4; VI 1.3, 2.1.4, 2.4, 3.2.2, 4.3-4.6, 5. The core of the book is the properties of various means. These means are listed in the Index both separately, such as Arithmetic mean, and collectively, Means. The simplest means are two variable means such as the classical arithmetic mean, ^(x -f y) and geometric mean, л/xy, and the less well known logarithmic mean, (y - x)/(logy- logs); see II 1.1, 1.2. 5.5, VI 2, 3. This leads to the question — what properties should a function f(x,y) have for it to be considered as a mean? Clearly we require / to be continuous, positive, defined for all positive values of both variables; a little less obvious are the properties of symmetry, f(x,y) = f{y,x), reflexivity, f(x,x) — x, homogeneity, f(\x,Xy) = \f{x,y), and monotonicity, χ < χ1 ,y < у' => f(x,y) < f{x',y')\ finally there is the crucial property of internality that justifies the very name of mean, mm{x,y} < f{x,y) < max{x,y}. Most of the means introduced are easily defined for η-tuples, η > 2, when these various properties are suitably extended; see II 1.1 Theorem 2, 1.2 Theorem 6, III Theorem 2(e) and VI 6. Of course there are many other properties of means that have been identified as of interest; these are listed in the Index under Mean Properties. The inequalities between the various means defined form the core material of the book. Again the two variable cases are the easiest II 2.2.1, 2.2.2, 5.5, VI 2, 3.1, 3.2.1. The fundamental result is the inequality between the arithmetic and geometric means, (GA), discussed in detail in II 2.4 where well over 70 proofs are given or mentioned; most are extremely elementary. The next basic result generalizes this and is the inequality between the power means, (r;s), HI 3.1.1. Integral forms of these results are also give; VI 1.2.2. Prom Notations 9 we see that every inequality between means of η-tuples can be regarded as saying a certain function of η is non-negative. For instance (GA) im- xxvi

plies that R(n) — n(2in — <6n) > 0. A stronger property of this function of η would be to show that it increases; stronger because -R(l) = 0; a similar discussion occurs for related functions that are not less than 1, (2tn/^n)n > 1 for example. This leads to the so called Rado-Popoviciu type extensions of the original inequality. Such are discussed for (GA) in II 3.1; the analogous discussion for (r;s) in III 3.2 is much more technical as the simplicity of the (GA) case has been lost. Finally many well-known inequalities arise from the discussion of mean inequalities— in particular the inequalities of Cauchy, (C), Holder, (H), Minkowski, (M), Cebisev and the triangle inequality, (T); see II 5.3, III 2.1, 2.2, 2.4. xxvn

a

I INTRODUCTION In this chapter we will collect some results and concepts used in the main body of the text. There is no intention of being exhaustive in any of the topics discussed and often the reader will be referred to other sources for proofs and full details. 1 Properties of Polynomials Simple properties of polynomials can be used to deduce some of the basic inequalities to be discussed in this book. In addition certain simple inequalities, needed at various places, are most easily deduced from the properties of some special polynomials. These results are collected together in this section for ease of reference. Convention In this section, unless otherwise specified, all polynomials will have real coefficients. 1.1 Some Basic Results The results given here are standard and proofs are easily available in the literature; see for instance [CE pp.420,1573; DI p.70; EMS p.59; EM8 p. 175), [Uspensky]. Theorem 1 A polynomial of degree η has η complex zeros, and if η is odd at least one zero is real. Theorem 2 [Descartes' Rule of Signs] A polynomial cannot have more positive zeros than there are variations of signs in its sequence of coefficients. Theorem 3 [Rolle's Theorem] If ρ is a polynomial then p' has at least one zero between any two distinct real zeros of p. Theorem 4 [Intermediate Value Theorem] A polynomial always has a zero between any two numbers at which its values are of opposite sign. Theorem 5 A zero of a polynomial is a zero of its derivative if and only if it is a multiple zero. 1

2 Chapter I The following result is basic to a variety of applications; see [BB p. 11; HLP ρρ.104~ 105], [Milovanovic, Mitrinovic & Rassias pp.70-71; Newton], [Dunkel 1908/9; Kellogg; Maclaurin; Sylvester]. The present form is that given in [HLP pp. 104~ 105]. Theorem 6 If f(x,y) = Σ™=0 с%хг уп~г has, as a function ofy/x, all of its zeros real then the same is true of all polynomials, not all of whose coefficients are zero, derived from f by partial differentiations with respect to χ or у. Further if a zero of one of these derived polynomials has multiplicity k,k > 1, then it is a zero of multiplicity к -f 1 of the polynomial from which it was obtained by differentiation. D The proof is immediate by repeated applications of Theorems 1,3,4 and 5. D Corollary 7 Ifn>2, and Π Π / ч p{x) = ς ^χί = Σ (\)άίχί w is a polynomial of degree η with c0 ^ 0 and all zeros real, then if 1 < m < k+m < η the polynomial q(x) = Σ™=0 (™)dfc+iX* has all of its zeros real. D Let f{x, y) = YZ=Q (^)а{х'уп-\ We are given that d0 φ 0, so (0, у), у ф О, is not a zero of /. Hence, by Theorem 6, (0,y),y φ 0, is not a multiple zero of any derived polynomial. This implies that no two consecutive coefficients of / can vanish. Save for a numerical factor дп~ш f jдкхдп-к~шу is equal to Σ™=0 (7)c4+^w~*, which by the previous remark does not have all of its coefficients zero. Hence the result is an immediate consequence of Theorem 6. D Remark (i) It follows from the above proof that if ρ is a polynomial of degree n, as in (1), with cq φ 0, and if for some k, 2 < к < η — 1, c& = c^_i = 0, then ρ has a complex, non-real, zero; [Wagner C]. Corollary 8 If η > 2, and ρ a polynomial of degree η given by (1) with c0 φ 0 and all of its zeros real, then for k,l < к <n — 1, d\ >dfc_idfc+i, (2) c\ >Cfc_iCfc+i. (3) The inequalities (2) are strict unless all the zeros are equal. D By Corollary 7 the roots of all the equations dk-i + 2dkx 4- dk+ix2 = 0, 1 < к < η - 1, (4)

Means and Their Inequalities 3 are real; from this (2) is immediate. Now from (2) and the definition of t4, 1 < к < η, see (1), 2 ^ (fc + l)(n-fc+l) cfc > τ-;——ττ Cfc-iQc+i > cfc-icfc+i, unless possibly either cj^-i = 0, or c^+i = 0; but then, by Remark (i), c& =£ О; and so in all cases (3) is proved. Finally, if for any к there is equality in (2) the associated quadratic equation (4) has a double root, and so, by Theorem 6, the original polynomial ρ of (1), has a single zero of multiplicity n. D Remark (ii) Inequality (2) is sometimes called Newton's inequality and will reappear later, II 2.4 proof (гж), V 2 Theorem 1; [DI p.192]. A direct proof can be found in [Nowicki 2001]. A converse to this inequality has been given; [Whiteley 1969]. Remark (iii) The above implies that if for some k,l < к < n— 1, c2, < Cfc_iCfc+i, then the above polynomial ρ has a non-real zero. Remark (iv) By writing (2) in the form d2^ > <^_ι<2£+ι, l<r<k<n — 1, multiplying, and assuming that cn = 1, we get the important inequalities, see V 2(6); <С\ > <_r_i; (5) similarly, cn_r > cn_r_1. (6) Example (i) A very important case of (1) occurs when — αϊ,...,— αη are the real distinct zeros of p, when p(x) = П^=1(х 4- α&); then cn = 1 and Сп_& = if Σΐ!αΰ ·-.aifc, 1 < fc < η; in particular cn_i = Σ^=1αΐ,θ3 = ΠΓ=ια*' see v 1(7). 1.2 Some Special Polynomials (a) [Dip.207], [Bullen 1996a; Cakalov 1963]. Define the polynomial pn,n > 1, by pn(x) = xn+1 - (п+1)ж + п. In particular then p\{x) = {x — l)2 > 0, ж =£ 1. Clearly χ = 1 is a zero of both pn and of p^. So, by 1.1 Theorem 5, χ = 1 is a double zero of pn. [This can also be seen from the identity pn(x) = (x~ l)2 Х]Г=о(п~^)жг·] This, by 1.1 Theorem 2, implies that ж = 1 is the only positive zero of pn.

4 Chapter I Since pn(0) = η > 0, and pn(n) = nn+1 — n2 > 0 if η > 1, we have by 1.1 Theorem 4, and the special case above that: if x>0, x^l, then rrn+1 > (n + I) x - η. (7) (b) [DI p.207]. Define the polynomial qn,n > 1, by gn(x) = (x + n)n+1-(n + l)n+1x. Then using an argument similar to that in (a), x = 1 is the only positive zero of qn. Hence if x>0, хф\, then [χ + n)n+1 > (η + l)n+1x. (8) 2 Elementary Inequalities In this section we collect some important elementary inequalities. 2.1 Bernoulli's Inequality Inequality 1.2 (8) leads to one of the basic elementary inequalities. Take the (n 4- l)-th root of both sides in 1.2 (8) and put η 4-1 = -, and у = ж — 1 г to get: if 2/>-1,2/^0, then (1 + y)r < 1 + r^; (1) This inequality (1), or (~1), can be shown to hold for arbitrary exponents r; [AI p.34; HLP p.40], [Herman, Кисета & Simsa p.109]. Theorem 1 [Bernoulli's Inequality] If χ > — Ι, χ φ 0, and if 0 < а < 1 then (1 4- x)a < 1 4- αχ- (Β) if either а > 1 or а < 0; when of course χ Φ — 1, then (~ B) holds. D (г) 0 < а < 1 Let f{x) = 1 4- αχ - (1 4- х)а, х > -1. Then /7(ж) = α(ΐ - (1 + ж)а-1)· Hence /(0) = /'(0) = 0 and it is easy to see that this point is the minimum of /. Alternatively we can look at f"{x) = a(l — a)(l 4- x)a~2, which shows that / is strictly convex, see 4.1 below for a definition. Either argument shows that f(x) > 0,ж φ 0, which is just (B). (гг) a < 0 or a > 1 The argument in (г) shows that (~B) holds.

Means and Their Inequalities 5 (Hi) Alternatively we can show that the result for a < 0 follows from that for a>0 If 1 + ax < 0 there is nothing to prove. So suppose that a < 0 and 1 4- ax > 0 and choose β so that β > 1 and 0 < -α/β < 1. Then from case (г) (1 + χ)~α/β < a 1 - -qX, or a Hence (1 + x)a > (1 + ^ж)^ > 1 + ax, by case (гг). D Remark (i) This important result is called Bernoulli's inequality, although the original inequality of this name is (~B) in the case а = 2, 3,...; [СЕ p.Ill; DI pp. 28-29]. It is still the subject of study; see [MPF pp.65-81], [Alzer 1990a,1991c; Mitrinovic L· Pecaric 1990a]. Remark (ii) (B) is equivalent to many of the fundamental inequalities to be discussed; see for instance II 2.2.2 Remark (i), III 3.1.2 Theorem 6. Remark (iii) The history of Bernoulli's inequality is discussed in [Mitrinovic L· Pecaric 1993]. Remark (iv) Proofs of (B) that differ from the one above are given later: see below 4.1 Example(iii), and II 2.2.2 Remark(i). Remark (v) (B) can be extended to ПГ=(1 + Xi) > 1 + ΣΓ=ι ж*' — 1 < ж < 1, χ φ 0; see [ΑΙ ρ.35; DI ρ.29; HLP ρ.60] and 3.3 Corollary 19(b). For further generalizations see II 2.5.3 Theorem 22 and [Pecaric 1983a]. (B) is given a symmetric form in the following corollary. Corollary 2 lfa,b>0,a^=b and 0 < α < 1 then ааа-г(Ь -а)>Ъа-аа > аЪа-г(Ъ - а), (2) т <аЪ+ (1 -а)а; (3) а if а > 1 or а < 0 then (~2) and (~3) hold. □ Substitute x = 1 in (B) to give the left inequality in (2); interchange a α and b to get the right inequality. D

6 Chapter I Remark (vi) A direct proof of (2) can be given using the mean-value theorem of differentiation1; [Ruthing 1984} Remark (vii) For a refinement of an integral form of (B) see [Alzer 1991c]. It is possible to obtain more precise inequalities; [HLP p.41]· Theorem 3 If χ > 1, and 0 < r < 1, or if χ < 1 and r > 1 then 1 (T_1\2 rr — 1 1 £(! - ryJ^T- < (^ - !) - V" < 2(1 ~~ γ)χ(χ _ 1)2· (4) If χ < 1 and 0 < r < 1 or if χ > 1 and r > 1 then (~4) holds . D These inequalities follow using a calculus argument similar to that used to prove (B). D Corollary 4 (1 + x)r =1 4- rx + 0(x2), as ж -» 0; (5) (1 + 0(r2))1/r =1 4- 0(r), as r -> 0. (6) D These follow easily from (4); [note that if χ > 0, then жг = 1 -f O(r) as r->0.] D Remark (viii) An alternative sharpening of (B) can be found in [Haber]. 2.2 Inequalities Involving Some Elementary Functions (a) the exponential Function [DI pp.81-83] If χ ^ e then ex>xe. (7) To see this note that у = x/e is a tangent to the curve у = log ж at the point (e, 1). Hence by the strict concavity of the log function, see 4.1 Corollary 3 below, χ — > logic, ж/e, which is equivalent to (7). e Remark (i) The graphs of ex and xe touch at (e,ee). In general the graphs of ax and xa, a > 0, cross at the two solutions of the equation ax = xa, χ = α, and The mean-value theorem of differentiation, sometimes named after Lagrange, states: If f is continuous on [a,b], differentiable on]a,b[ then there is a c, a<c<b, such that f'(c)= ъ^а ' t*-^ p. 1153]. We call such a point с a mean-value point of f. Further note that the mean-value theorem of differentiation can be used to prove that: if /'>0 and every sub-interval contains a point at which f'>0, in particular if /'>0 with f'(x)=0 at only a finite number of points, then / is strictly increasing.

Means and Their Inequalities 7 x = a, say. If a = e,a = a , if 1 < a < e then a > e, while if α > e, then о 1 < α < e; for α < 1 take α = oo. If / = [a, a], or [a, a], then ax < xa if χ el and a* > xa if ж £ J; [Bullen 2000; Qi Sz Debnath 2000b; Smirnov]. If χ φ 0 then еж>1 + ж. (8) This follows in several ways: (г) the strict convexity of the function ex implies by 4.1 Corollary 3 that at each point its graph lies above the tangent, and у = χ -f 1 is the tangent to the graph at χ = 0; χ2 (гг) by Taylor's theorem2 since ex = 1 4- x 4- "тг^ f°r some j/ between 0 and x\ (%%%) note that f(x) = ex — 1 — χ has a unique maximum at χ = 0. Remark (ii) Of course the Taylor's theorem argument can be used to prove that xn ex > 1 4- ж 4- ■ · ■ Η -, ж > 0. In particular, taking n = 2 and replacing χ by χ — 1 η! we get ex~1/x > -(ж 4- l/ж), x > 1. Another well-known fact is that (l + i)n<e<(l + I)n+1, η EN*. 4 TV V TV It can be shown that the left-hand side increases to e, and the right-hand side decreases to e, as η —> oo; see II 2.5.3(£). A simple deduction from these inequalities is a crude estimate for the factorial function η!, η Ε Ν*; [Klambauer p.410]. nn nn+i τ < n\ < on—1 More generally, [Dip.81], [Wang С L 1989a], if χ φ 0, ;(1 + -Ρ<(1 + -)"<Λ neN', which shows that limn_>oo (l Η—) = еж. 4 τν (b) The Logarithmic Function [DI pp. 159-160} If ж > 0,# =£ 1 then ж — 1 < log ж < χ - 1. (9) Taylor's theorem states:if / has n,n>0, continuous derivatives on [a,b], and if /(n + 1) exists on ]a,b[ then /(χ)=Γη+ι(/;α;χ) + βη+ι(/;α;χ) where T^iC/jajx)-^^ /(^,(α) (x-a)fc is the Taylor polynomial of order n+1 centred at a, and fiTt+i(/;a;x)=(n_114, J (ж—i)n/^ + 1) (i) at is the (n+l)-st Taylor remainder centred at a; also for some c,a<c<b, Rn + i(f;a;x)=f^n+1^ (c) ^T^K^ · The expression Tn^1+Rn + i is called the Taylor expansion of f. [EM9 pp.1355-136].

8 Chapter I fx 1 The right-hand inequality is immediate because log χ = - at < χ—1; or because Λ t if f(x) = 1 — ж + log χ then / has a unique maximum at χ = 1. In addition a simple change of variable changes (8) into the right-hand inequality in (9). The left-hand inequality follows by a simple change of variable in the right-hand inequality. Remark (iii) For another amusing way of proving (9) see 4.5.3 Example (iv). A useful inequality is the following, [Wang С L 1989a}; if χ φ 1 then log ж < n(x1/n - 1) < x1/n\ogx, η e N*, which shows that Нп^-^ n(rc1/'n — 1) = log ж. If χ > — 1 then 21x1 <|lOg(l+a0|<-JfL. (10) 2 + x bv y/l+x' *2χ χ Consider the functions f(x) = log(14-х) — 7: , and д(χ) = log(l 4-x) — 2 + a;'—^-^t*/ νΓΤΈ- Then simple calculations show that /(0) = g(0) = 0 and gf < 0 < /;, which implies (10). Remark (iv) For other inequalities involving the logarithmic function see II 5.5 Corollary 13 and Remark (ii), and [Mitrinovic 1968b]. (c) The Binomial Function [DIpp. 35-37] Another useful inequality is the following. Theorem 5 If either (a) x > 0 and 0<q<p,or(b)0<q<p and —q<x<0, or (c) q < ρ < 0 and —p>x>0 then (i + -)g<(i + -)p. (и) q ρ Inequality (~11) holds if either (d) q < 0 < ρ and q > χ > 0; or (e) q < 0 < ρ and -ρ < χ < 0. D For a simple proof by Bush see [AIp.365], [Bush]. D This inequality is a generalization of (B); for a special case see 3.3 (20), and II 2.5.3(5). In addition there is: 'n\ к {l + x)n>l + nx + --.+ lkjx\ (12) provided χ > 0,0 < к < n,n ^ I. Inequality (12) and the inequality in Remark (ii) are special cases of Gerber's inequality, [DIpp.101-102], [Alio, Bullen, Pecaric к Volenec 1998; Alzer 1990a; Gerber 1968].

Means and Their Inequalities 9 (d) The Circular and Hyperbolic Functions [DI pp. 131-132, 250-252] Sin Ж 7Γ cos ж < < 1, x φ 0; sin ж < χ < tan ж, 0 < χ < —; (13) χ 2 sinn x cosh ж > > 1, ж φ 0; tanha; < ж < втпж, ж > 0. (14) ж Remark (v) For an extension of the right-hand set of inequalities see II 5.5 Remark (vi). Further the left-hand side of the first set of inequalities in (14) can be improved to (coshж/3)3, see VI 2.1.1 Remark (x). (e) Maximum and Minimum Function Inequalities [DI pp.134] If Ql and k are real η-tuples then maxa -f minb < max(a -f b) < max a + maxb, (15) maxaminb < maxab < maxamaxb. (16) The right-hand inequalities are strict unless for some i, a* = maxa and bi = maxb. Similar inequalities hold for sequences or infinite sets of real numbers but in that case we usually write inf and sup for min and max respectively. (f) A Simple Rational Function Let a, b > 0 and put Дж) = - + -, ж>0. α ж Noting that fix) = —«(ж2 — ab) we see that / has a unique minimum at ж = αχΔ жо = Vab and that f(xo) = у о = 2у/Ь/а. In particular / is strictly convex, strictly decreasing on the interval ]0,жо], and strictly increasing on the interval [жо,оо[. Alternatively the equation fix) = у has two roots, Ж1,Ж2 say, and Ж1Ж2 = ab, Ж1 + X2 = ay. So χ b [b , ч - + ->2W-, 17 а ж V а with equality if and only if ж = yah] further if ж > Ж3 > yab then ж b Ж3 b , s - + ->— + —, 18 а ж а жз with equality if and only if ж = жз; and finally if χγ < χ < X2 then

10 Chapter I with equality on the left if and only if ж = xo, and on the right if and only if X = Χι, ОГ X = X2- Some special cases are of some interest. (i) If a = b = 1 inequality (17) gives the familiar x + ->2, x^l. (20) χ A direct proof follows immediately on noting that for χ =£ 1 (20) is equivalent to (x - l)2 > 0, or by noting that (x - 1)(1 - l/x) > 0. (ii) If α = — < 1, Ь = 1 then x0 = —^ < 1 So taking ж3 = 1 in (18) we get that if χ > 1 then аж-f-- > a-f-1, (21) χ with equality if and only if χ = 1. This is an inequality due to Chong, [Chong К Μ 1983], A direct proof can be given as follows: ax -\— =(a — l)x 4- (x Η—) χ χ >(α-1)χ + 2, by (20), >α + 1. For equality at the first step we need χ = 1, and then there is equality at the second step. Inequality (21) can be written in an equivalent form: put a = u/v, и > ν > 0 then ux + ->u + v, (22) χ with equality if and only if χ = 1. (iii) Given ra, M, 0 < m < Μ choose Χγ = га, Х2 = Μ and a = b = y/mM then from (19) ос у/^Ш [М fm _ ra + M yfmM x Vra V^ VmM with equality in the left inequality if and only if χ = y/mM, and in the right inequality if and only if χ = m or χ = Μ. Inequality (20) has been generalized by Korovkin; [Korovkin p.8]. Lemma 6 If x^ is a positive η-tuple, η > 2, then ii + ^ + ... + Ξϋτ! + ^>η, (24) X2 %3 xn Xl

Means and Their Inequalities 11 with equality if and only if χ is constant. Π The proof is by induction, noting that the case η = 2 is just (20) with α = b = X2 and x = x\- So assume (24) holds for integers less than n, and that x = minx. Then the left-hand side of (24) is equal to X\ %n—l . ί Xn — 1 Xn Xn —l\ X2 Xl V Xn Χγ Χγ ) >(n — 1) + ( n~ Η— ^—— J, by the induction hypothesis, \ Xn X\ X\ J {Xn ~ Xn— l)\Xn ~~ Xl) . _n _j_ _ > n. XlXn The case of equality is immediate. D Remark (vi) For further generalizations see II 2.4.5 Lemma 19 and II 2.5.3 (φ) Theorem 25. 3 Properties of Sequences 3.1 Convexity and Bounded Variation of Sequences Let α = (αι,α,2,...) be a real sequence and define the sequences Ака, к e N*, by recursion as follows:3 Αλαη = Δαη = an -an+b η e N*; Afcan = A(Afc-1an), neN*,A;>2. Then it easily follows that Afcan = ^(-lW . αη_μ, η Ε Ν*, and that if 1 < j < k, Δίβ» = Σ(<ί*ϊ-Τ2)Δ*α'+"-1· (2) Convention Throughout this section we assume that: ~ ] = 1, and, if η > 2, that (П~ ) = 0· (1) \ 7. / г=0 We will also use the notation Aka, where Δαη=-Δαη = αη+ι-αη, Akan=AyAk 1ατι), η€Ν*, fc>2 So Afcan = (-l)fcAan, fc,n€N*

12 Chapter I Definition 1 (a) A sequence α is said to be к -convex, к e N*, if the sequence Aka is non-negative. (b) A sequence α is said to be of bounded k-variation, к е N*, if £('+*-')i^i<~ i—i v ' Example (i) A sequence that is 1-convex is exactly a decreasing sequence; a 2- convex sequence, usually called a convex sequence, has an — 2an+14-an+2 > 0, η > 1. Example (ii) A sequence a is of 1-bounded variation, or just bounded variation, if Σ£ι |α* — fli+i| < со. Clearly the sequence of partial sums of a series is of bounded variation if and only if the series is absolutely convergent. Remark (i) If / is a convex function, see 4.1 Definition 1, then an = f(n),n = 1,... is a convex sequence; more generally if / is /c-convex, see 4.7, then an = (-l)nf(n),n= 1,... is /c-convex. [DI pp.6^-65, 191, EM2 p.419]. Remark (ii) If the sequence —a is convex, /c-convex we say that a is concave, k-concave. Remark (iii) The definition of a k-convex η-tuple, η > к, is immediate. The partial sums of a series can be written as the difference of the partial sums of two positive termed series, Bn = Y%=1 bn = Σ™=ι Κ ~ ΣΓ=ι Κ = pn - Nn; and the series Σ bn is absolutely convergent if and only if both of the series Σ7=ι ^η and Σ7=ι ^η converge. This can be expressed in terms of the present concepts as: if a sequence is bounded then it is of bounded variation if and only if it can be written as the difference of two bounded decreasing sequences. The main result of this section, Theorem 3 below, is a generalization of this result due to Dawson, [Dawson]. First we prove a basic lemma Lemma 2 If α is a sequence that is bounded and k-convex, respectively of bounded k-variation, к > 2, then: (a) α is p-convex, respectively bounded p-variation, 1 < ρ < к — 1; (b) Hm (n + J~lS)Ajan=0, 1 < j < к - 1; 7WOO \ j J (c) У| . ., )Ajai =ai - Hm an, 1 < j < k. Δ—' V j — 1 J n—>oo

Means and Their Inequalities 13 Ρ uy k = 2 and α is bounded and convex. This implies that Δα is decreasing and so limn^oo Δαη exists. Further the assumption that α is bounded implies that this limit is zero; hence Δα > 0. This is (a) in this case. Now η n-l αϊ ~ αη+1 = Σ Δα* = Σ ^2аг + ηΔαη, (3) and noting that Δα > 0, Δ2α > 0, and that α is bounded, the series in (3) converges, which implies (c). Hence limn^oo an exists and is finite, which from (3) and the convergence of the series shows that limn^oo nAan exists and is non-negative, and so is zero. This gives (b) and completes the proof of this case. (и): k = 2 and α is bounded and of bounded 2-variation. From (3) the sequence nAan,n e N*, is bounded and hence Σί^ι |Δα^| converges, which is (a) in this case. Further limn^oo &n exists, and so from (3) limn_>oo nAan exists, A say, and assume that Α φ 0. Then for some n0Gf, 2 A i=no 1 Δα^+ι Acii oo < Σ г\А<ц\ Δα,· 1 г=п0 oo = £ г|ДЧ < oo. A Hence TT (—-r^—) converges absolutely to some non-zero limit. This contradicts г=по the convergence of ΣΖι \^cii\, and so A = 0. The lemma in this case now follows from (3). (iii): к > 2 and α is bounded and /c-convex. Since Δ^α = Δ2(Δ^~2α) the sequence Δ^~2α is convex and bounded. Hence (a), in this case, follows by induction. In particular α is convex and so (b) and (c) hold with ,7 = 1, and j = 1,2 respectively. Suppose that I < jo < к and that (c) holds for j = jo- Then since this implies ff'^0"1) Δ^'0+1α, < oo, and lim (П + j° ~ Λ Δ^0+1αη exists, ^ V ^0 J гь^оо \ jQ J A say; assume that Α φ 0.

14 Since Chapter I '"+rv-(tc;*T2))^^(":*ry- it follows that there is an no £ N* such that if η > no then 2n n\ jo J \ Jo-l J oo / · ι · o\ This contradicts the convergence of ^ ( ■ ι ) ^°ai' so we ^ave (c) *n ^e case к = Jq. Hence A = 0; that is (b) holds with j = j0 and so, from (4) and (c) with j = j0, V (г + *> - l) Дл+Ч = g Λ + Λ - 2λ дЛ+1 Ит that is to say we have (c) with j = jo + 1. This completes the proof of this case. (iv): к > 2 and a is bounded and of bounded k-variation. Afc-1an,n G N* converges to zero. Then for some n0 Ε Ν* and A > 0 we have that ^Σ vfc-l &i+l Δ*-1^ < Σ г + к - 2 fc-1 ΙΔ^α 1- Δ^α -г + 1 Δ*-1^ - ς С;*;2 V-i Hence by arguing as in (гг), lim Ак~1ап = A^Q. n-+oo Since a is of bounded /c-variation, and (5) f>2|A^| < Σ (* + * χ 2V«i| < oo, we have that Ak~2a is of bounded 2-variation. So, by the case к = 2, Ak~2a is of bounded variation; that is Υ^λ \A(Ak~2ai)\ = Y^°Z1 |Δ^-1α^| < oo, which contradicts (5). Hence there is a strictly increasing

Means and Their Inequalities 15 sequence щ e N*,z Ε Ν*, such that lim ( )Ak~1ani = 0, which implies that for some M,l*k_l J ΙΔ*"1^ | < M. Now, П 7·—1 еГГлУ'-^е^У^Ч^-Г^'*· <"£'с+ду°<|+«· 1—1 Ч ' Since α is of bounded /c-variation this last inequality implies that a is also of bounded (k — l)-variation, and a simple induction completes the proof of (a) in this case. / γ) . _L_ iji ) \ The two series in (4) with j0 = к -1 converge, and lim ( * Afc-1ani = 0; г—юо у /С — 1 у (η 4- к — 2\ )Afc~1an = 0, and (b) follows by induction, /c — 1 / Finally, from (4) with j0 = к - 1, JT Г + * ~ 2 J Δ4 = ]Γ Γ + * ~ 3 J Δ*"1^, г—ι and (c) follows by induction, and (5) is already proved when к = 2. D We can now prove the main result of this section. Theorem 3 If a is bounded then it is of bounded k-variation if and only if it is the difference of two bounded k-convex sequences D We may assume к > 1 as the case к = 1 is well known, see the comments preceding Lemma 2. (г) Assume that a= b — c, where b and с are bounded and /c-convex. Then, г—ι х ' i=i ч / г—ι ν / by Lemma 2(c) with j = к. (ή) Assume that α is bounded and of bounded /c-variation. Then by Lemma 2(c) limn^oo an exists, A say. Prom a simple extension of Lemma 2(c), Y~] ( , Afca^+n = an+i — A. Now define b"+i = E('t-^2)|A4+nl'neN'

16 Chapter I Then b is bounded and ^ = Σ С t -Τ2) ιδ*—ι - Σ С1 -12) |дк— HA'a^l + f^ + ^Wa +n-l| i+n-l\ \ К — Δ J i—2 i=i ^ ' so, by an obvious induction, Δ^6η = £ f "t * ~ Ϊ7 2) ΙΔ^+η-ιΙ > 0, 1 < j < λ; ΐ=ι \ J ' showing that Ь is /c-convex. Now define c = b — a when, using (2),we have for 1 < j < к that, Ajcn=Ajbn -Ajan This completes the proof. □ Remark (iv) More results on convex sequences can be found in the following reference, where further references are given; [Pecaric, Mesihovic, Milovanovic &: Stojanovic]. 3.2 Log-convexity of Sequences Definition 4 Given two real sequences α = (αο, αϊ,...) and b= (bo, bi, · ■ ·) ^ne sequence c = a*b denned by η Cn=^2 arbn-r, П в N, (6) r=0 or by the product of the formal power series Σ °rxr = ί ΣατχΤ) (ΣbrxT I' (7)

Means and Their Inequalities 17 is called the convolution of α and b. Definition 5 (a) If 0 < а < со then the positive sequence c= (со, ci,...) is said to be a-logarithmically convex, or just α-log-convex, if for η e N*, f /α4-η-1\ /n-f 1 , .- , Cn+iCn-l, II OL ψ OO, cl<< Cn+iCn-i, it α = oo. η When а = 1 we will just say log-convex, and when а = со we will say weakly log-convex. (b) If inequality (~8) holds then we say that the sequence is a-log- concave; or just log-concave if а = 1; if а = oo we say the sequence is strongly log-concave. [DI p. 163]. Remark (i) In the convex case the smaller α the stronger the condition to be satisfied, while in the concave case the larger α the stronger the condition. Remark (ii) The sequence с is log-convex if and only if c* <cn+1cn_b neN*; (9) equivalently if and only if -<-<··■< ^~ < .. · < Cm+n < · · ■, n, m £ N*; (10) Co ci cm+i cm+n+i in the case of log-concavity the inequalities (~9) and (~10) hold. Example (i) The sequence α defined by an = (-ψ(-α) = «(«+l)-(« + "-l)> n 6 N^ (11) \ η ) n\ is o/-log-convex, respectively concave, if a' > a, respectively a' < a. Example (ii) The sequence bn = —, η e N*, is 0-log-concave and so α-log-convex for all α > 0. Example (iii) The sequence cn = —, η £ Ν*, is weakly log-convex and so a-log- convex for all a < oo. Example (iv) Prom 1 Corollary 8 we see that if a polynomial has all of its roots real then its coefficients are log-concave; in particular {(^), 0 < к < η} is log- concave.

18 Chapter I Remark (iii) It is useful to notice that с is α-log-convex, respectively concave, if and only if с is log-convex, respectively concave, where for η e Ν* , cn = c/an if 0 < a < со, cn = ncn if a = 0, and Cn = n\cn if a = со; where an is defined in (11). The results of this section show that the property of log-concavity is preserved under the operation of convolution. Not all the results are needed in the sequel but are included for the sake of completeness. Theorem 6 If a and b are log-convex, respectively log-concave, so is с where °η = Σ r=0 Οι'ρΟγι — γ>, Tl t (12) D (i)' α and b are log-convex; [Davenport L· Poly a]. The proof of inequalities (9) is by induction on n. The case η = 1, c0C2 — c\ = a^ofo — b\) + bo(ao«2 —a?) > 0, follows from the log-convexity of a and Ъ. Now assume that (9) holds for all η, 1 < η < к — 1. Then, noting that ( I — ( 1 + ( ) > we can write c^ = c'k_1 + 4'-i where the sequences d, c" are defined using (12) with the sequences a/, b, and a, b" respectively; where a' = a'0 = (αϊ, аг,...) , έ" = b[) = (bi, 62,...). Hence, cfc = (Cfc-l + Cfc_!) = Cfc_! + 2cfc_1Cfc_1 + Ск_г <ck_2ck + 2л/ ck_2ckck_2ck 4- ck_2ck, by the induction hypothesis , <4-24 + 4-24' + 4-24 + 4-24, by (GA), see И 2.2.1 (2), =Cfc_iCfc + i, as had to be proved. (гг): а and Ь are log-concave; [Wbiteley 1962b, 1965]. η—ι / _ ^n Rewrite definition (12) as: cn = V^ ( So: (ar_|_ibn_r_i -f arbn-r). ^n *-Vi — lQi + l r ^ Σ Lr=0 аГЬи = A 4-В. Σ r=0 η —ι n-1 Σ L r=0 r n-1 T" CLfOn — r J CLr0ri — r / , ( ) («r+ibn-r 4- arb, Lr=0 'n-r+lj

Means and Their Inequalities 19 where A = 2^f (^s^r+s+l — br+sbs-i) ( ( _ 1 ) ( _i_ _ ι )an-r-s+l«n r+s<~n+l η \f n-1 \ s — 1/ \r 4- s — 1/ and Б is obtained from A by interchanging a and b. The log-concavity of b shows that the first bracket in A is non-negative, (10); further, since (r _^_ J ("l J) > (s ! x) (r+^ ^ the second bracket in A exceeds ί _ , J ( , _ , ) {^n-r-s+iCLn-s - an-s+i^n-r+s), which is non- negative by the log-concavity of a, (10). A similar argument shows that В is non-negative and completes the proof. D Remark (iv) It is not difficult to see that there is equality in inequalities (9) for c, defined by (12), if and only if equality occurs in all the corresponding inequalities for a and b. Noting that if dn = cnxn, η e N, then с and d are a-log-concave together leads to the following generalization of Theorem 6. Corollary 7 If If a and b are log-convex, respectively log-concave, so is c{x,y) where cn{x,y) = Σ \Z\arbn_rxryn r,n βΝ,χ > 0, у > 0. r=0 ^ ' (13) Theorem 8 (a) If If α and b are log-concave, so is a-kb. (b) If If a is a-log-convex, and b is (1 — a)-log-convex, 0 < a < 1, then a-kb is log-convex. D (a) [Menon 1969a] If с = а • b then, П 71—1 71 + 1 cn ~ cn-lCn + l =( 2_^ ar^n-r) — ( 2_^ arbn-r-l) ( 2_^ arbn-r+l) r=0 r=0 r=0 71—1 71 71 — 1 71 = (2^arbn_r)(^arbn_r) ~ ( /J arbn-r~i) ( /J arK-r+i) r—0 r=0 r=0 r—0 71 71—1 + anb0(2^arbn_r) - an+ib0(2_^arbn_r_i) r—0 r—0 = A + В + С,

20 Chapter I say, where 71 — 1 П П—1 П A = ^2^2arak{bn-rbn-k -K-r-ibn-k-^-i) =ΣΣάτ>]<> say' r=0 fc=l r=0 fc=l n-1 5 = ^ arao(bn_rbn — bn_r_ibn+i), r=0 η η—ι С =апЬо ^^ arbn-r — αη+ι&ο ^, arbn_r_i r=0 г—О η =апЬпа0Ьо + Ьо 2_^ bn-r (о,паг — an+iar_i). Г=1 jB and C are easily seen to be non-negative by the log-concavity of a and b. Since, in А, dr,r+i = 1 we get on combining dr-i^ and dfc_i>r that, 71 ^ = Σ (dr-l,fc+dfc-l,r) T-,fc=l r<fc+l 71 r,k=l r<fc + l which is non-negative by (10). (b) [Davenport &, Polya] Using (11), Example (i) and Remark (iii), and letting β=1-α, η η Cn = у ^ CLrOn — r = / JO^rPn — r0jr^n — r· 1,14) r=0 r=0 Simple calculations from (11) lead to: fn\ (a + r - !)!(/? -f η - r - 1)! aA"r"W n!(a-l)!(/?-l)! "W (a - 1Ж/3 -1)!Уо ( } ^ Substituting in (14) and using the notation of (13) ^'(a-W-Dl/1^1-^'1^1-*^ <{а_щр_1у1 *Q_1(1 - i)'3" V^n-i(i, 1 - *)cn+1(i, 1 - i) di, by Corollary 7 and (9).

Means and Their Inequalities 21 A simple application of (C)-|, see VI 1.2.1 Theorem 3(b), gives (9) and completes the proof. □ Remark (v) In the proof of part (a) of this theorem С > 0, and so the inequalities (9), for c, are strict. In part (b) however these inequalities can be equalities, and are so if and only if all the inequalities for a, and for b are equalities. Remark (vi) The results in (b) is best possible in that the analogue of (a) is in general false; further the result does not hold in general if a = 1. The following examples illustrate this. Example (v) If α, β_ be two sequences defined as in (11); then α* β = a -f β and so if α4-β > 1 the convolution is not log-convex but only c/~log-convex, a' > α+β. Example (vi) Let an = —,bn = l,n e N*, then a is 0-log-convex, see Example (ii), and b is log-convex; but a*b = с where cn = Σ™=11/r, η e N*, which is not log-convex. 3.3 An Order Relation for Sequences Definition 9 An η χ η matrix is S = (sij)i<ij<n with non-negative entries is called doubly stochastic if η η 2^ Sij = 1, 1 < г < η, and Yj Sij = 1, 1 < j < n. (15) If in addition in any row or column all elements except one are zero the matrix is called a permutation matrix; equivalently a doubly stochastic matrix is called a permutation matrix if each row, and column, contains an element equal to 1. [CE pp.1349,1743; EM9 p.9; MO p.19} Example (i) If ^, q[ G R^ and qi -f q'i = 1, 1 < г < η, then the following matrices, all η χ η except the Qi, are doubly stochastic: /; ^J; Qi=(j. 41 ); Pi= Ι Ο Qi О |,1<г<п Pn= [ Ο /η_2 Ο ] ; P = ]JPi Definition 10 A real η-tuple b is called an average of the real η-tuple α if for some doubly stochastic matrix S, b = aS. Remark (i) To say that b is an average of α is a statement that does not depend on the order of the elements in either of the η-tuples, and is a transitive relation

22 Chapter I on the set of η-tuples; that is if a is an average of b, and b is an average of с then α is an average of c. Remark (ii) To say that b is an average of a is to say that each element of b is a weighted arithmetic mean of the elements of a; see II 1(3). Remark (iii) If b is an average of a and a is an average of b then b is a permutation of a; conversely b is a permutation of α then b is an average of α and a is an average of b. Remark (iv) If 1 < m < η and b = (Ь(1),Ь(2)) where Ь(1) = (bb...,bm) and k — (&т+ъ · · ·, bn) , with a = (а^\а^) defined analogously; then b^ an average of gSl\ i = 1, 2, implies that Ь is an average of a. Definition 11 (a) If а, Ь are decreasing real η-tuples we say that b precedes a, b-< a, if: (16) Bk <Ak, 1 < к < п. (b) Generally if a,b are real η-tuples we say that b -< α if when the two n-tuples are arranged in decreasing order they satisfy (16). Remark (v) This relation is an order relation on the set of decreasing real n- tuples, and a pre-order on the set of real n-tuples. Example (ii) If α is an η-tuple withai 4- ■ · ■ 4- an = 1, then (1/n,..., 1/n) -< a -< (1, 0,..., 0); use (16) and (1/n,..., 1/n) = a( — J), see Example (i). Example (iii) If с is a constant real η-tuple and b ·< a, respectively b is an average of a, then b 4- с -< a 4- c, respectively b 4- с is an average of a 4- c. Remark (vi) A full discussion of this concept can be found in [MO], and an introductory survey is given in the article [Marshall L· Olkin 1981]; see also [DI pp .198-199]. Lemma 12 b is an average of α if and only if it lies in the convex hull of the n! points obtained by permuting the elements of а D This follows from a theorem of Birkhoff; see [MO p. 19], [Marshall L· Olkin 1964]. [A definition of convex hull is given in 4.6 Definition 35(b) below]. D Lemma 13 [Muirhead's lemma] If b -< α then b = 0T1T2 ■ · ·Γη_ι, where Ti is an η χ η matrix of the form XI + (1 — X)Qi, where Qi is a permutation matrix that differs from I in just two rows, or equivalently just two columns, 1 < i < η — 1. D See [MO pp.21-22]. D

Means and Their Inequalities 23 Theorem 14 [Hardy, Littlewood L· Polya] A real η-tuple b is an average of α if and only ifb -< a. D See [HLP p. 49; MO p.22], [Rado]. D Remark (vii) If (16) is replaced by Bk < Ak, 1 < к < η, (17) we say that b weakly precedes a, b -<w a; [MO pp.9-11]. A simple but useful idea is given in the following definition. Definition 15 Two η-tuples a,b are said to be similarly ordered when, (dj - ак){Ьа - bfc) > 0, 1 < j, к<щ (18) if this inequality is always reversed then the η-tuples are said to be oppositely ordered. Remark (viii) Clearly α and b are similarly ordered if and only if a simultaneous permutation of them both leads to decreasing η-tuples; and they are oppositely ordered if and only if a simultaneous permutation leads to one increasing n-tuple and one decreasing η-tuple; see [AIp.10; HLP p.43; MO pp.139-140]. A very simple but useful result is the following; see [DI pp.222-223; HLP pp.261- 262], [Herman, Kucera & Simsa pp.145-148], [Vince]. Theorem 16 if two η-tuples a and b are given except in arrangement then the sum Σ aibi is greatest if they are similarly ordered and least if they are oppositely ordered. D It is sufficient to note that (djbj + акЬк) — (djbk + a^bj) = (clj — ak)(bj — b^); for full details see the references above. D Remark (ix) If any of the inequalities in (18) is strict then the resulting inequalities in Theorem 16 are also strict; so there is equality if and only if either a or b is constant. Corollary 17 If o/ is a rearrangement of the positive η-tuple α then ±аЛ>п. (19) г=1

24 Chapter I Further the inequality is strict unless g/ = a. D It is sufficient to remark that a and дГ1 are oppositely ordered and the right-hand side of (19) is just Σ™=ι (α*—)· ^^е case °^ ecLuamT is immediate. D Remark (x) This result of Chong Κ Μ, [Chong К М 1976с], has been generalized; see [Ioanoviciu]. The following theorem is a simple application of these concepts due to Chong К Μ; [Chong К М 1976с]. Theorem 18 ИЪ<а then Щ=1 bi > Щ=1α*· D Suppose that b = λα 4- (1 — λ) α', where a/ is a permutation of a. Then, Π Π / / \ пг-пН1-^) η—-л " 1—1 ν ' к < =χ;λ-Μ-λ)* Σ Πί fc=0 l<2i<-<2fc<nj=i 2j >Σλη-*(1-λ)*ί"), by Corollary 17, к=0 The general case follows using Lemma 13. D Corollary 19 (a) If χ > -1 then for n>2, n-l (20) (Ъ) If either a^ > 0 or — 1 < a^ < 0, 0 < г < η then η η 1 + £><П(1 + сц). (21) г=1 г=1 D (a) Use Theorem 18 and the fact that: (l 4- ж/п, 1 4- ж/п, •••,14- χ/η) -< (1 4- ж/(п - 1), 1 4- х/(η - 1), ■ · ■, 1 4- χ/{η - 1), 1). (b) Use Theorem 18 and the fact that (14-αχ,..., l4-an) -< (14-X]^ a», 1,..., l). D Remark (xi) (20) is a special case of 2.2(11).

Means and Their Inequalities 25 Remark (xii) The order relation discussed in this section has been extended to functions, see VI 1.3.4. 4 Convex Functions The concept of convexity is very important for this book. However this subject receives full treatment in several readily available sources and so this section will merely collect results. For further details and proofs the reader is referred to the following books: [AI pp. 10-22; HLP pp.70-83; RV; PPT], [Aumann & Haupt; Popoviciu], and for general information see [CE p.326; DIpp.61-64; EM2 pp. 415- 416]. The basic reference for all the topics of this section is [RV] where full proofs and many references can be found. As the topic is one of active research the more recent [PPT] contains much of this newer material as well as an extensive and up-to-date bibliography. f(y) (0<λ<1) (1-λ)ί(χ)+λ%) ί([1-λ>+ λν) Figure 1 Graph of convex function graph 0f f [1- λ]χ+λγ χ (λ<0) [1- λ]χ+λγ (0< λ< 1) У [1-λ]χ+λγ (λ>1) 4.1 Convex Functions of a Single Variable Definition 1 (a) Let I be an interval in R, then f : 11—► R is said to be convex, on J, if for all x,y e I, and for all λ, Ο < λ < 1, f(\x +T=\y) < Xf(x) + (1 - A)/(y). (1) If (1) is strict whenever χ ^ у and λ ^ 0,1, then f is said to be strictly convex, on J. (b) If in (a) inequality (1) is replaced by (~1) then f is said to be concave, strictly concave, on J. Remark (i) If / is both convex and concave then for some m and c, f{x) = тхЛ-с\ that is to say, / is аЖпе.

26 Chapter I Remark (ii) The graph of a (strictly) convex, concave, function is also said to be (strictly) convex, concave. Remark (iii) The simple geometric interpretation of (1) is that the graph of / lies below the chords of that graph; see Figure 1. Remark (iv) If Xi, 1 < г < 3, are three points of J with x\ < x2 < хз, then (1) is equivalent to: xs — x\ хз — x\ or more symmetrically, /Oi) + /Ы + /Ы > 0. ,2*ч (xi - χ2){χι - x3) (x2 - ^з)(^2 - xi) {хз - ^ι)(χ3 ~ Χ2) ~ or even 1 xx f(xi) 1 X<2 ffa) 1 хз /(я*) Lemma 2 If f is convex on I and if > 0. (2**) n(x,y) = f{y) f{x\x,yel,x^y y-x then η is increasing in both variables and strictly increasing if f is strictly convex. Π Another way of writing (2) is: /(*l} ' /Ы < f^ ~ /(Жз). So, if хг-х2 Х2-хз ^1,2/1,^2,2/2 e I with xi < yux2 < 2/2, f(x2) - f(xi) < /(2/2) - /(2/3) ,3x ^2 ~ Ж1 ~ 2/2-2/1 The lemma is an immediate consequence of (3). D Remark (v) In other words the slopes of chords to the graph of a convex function increase to the right; further these slopes increase strictly if / is strictly convex. This has some interesting implications. Corollary 3 (a) At any point on the graph of a convex function where there is a tangent the graph lies above the tangent. (b) If f is convex on I but not strictly convex then it is afRne on some closed sub-interval of I. Remark (vi) The property (a) is illustrated in Figure 1.

Means and Their Inequalities 27 Remark (vii) The intervals where a convex, not strictly convex, function is affine can be dense as the integral of the Cantor function4shows. Theorem 4 Let f be convex on I then: о (a) f is Lipschitz on any closed interval in I; /M f'± exist and are increasing on I and f'_ < /+; further if f is strictly convex these derivatives are strictly increasing; (c) f exists except on a countable set, on the complement of which it is continuous. (d) If f and д are convex then so is λ/ + μ# if λ, μ > 0. (e) If f and д are convex, increasing (decreasing) and non-negative then fg is also convex. if) If f is convex on I and g is convex in J, f[I] С J, g increasing, then д о / is convex on I. D See[RVpp.4-7,15-16}5 D Remark (viii) If / is an affine function then (f) holds without the assumption that д is increasing. Corollary 5 (a) f is convex, respectively strictly convex, on ]a, b[ if and only if for each xo, а < xo < b, there is an affine function S(x) = f(xo) + m(x — x0) such that S(x) < f(x), а < χ <b, respectively S(x) < f(x), а < χ < b, χ ^ x0. (b) If f is differentiable and strictly convex on an interval I and if for с е I we have f'(c) = 0 then /(c) is the minimum value of f on I and this minimum is unique D (a) If / is convex, respectively strictly convex, take in S any value of m such that m G [fL(xo),f+(xo)]· Conversely suppose that such an S exists at xq and let xo = \x 4- (1 — X)y where x, у e]a, b[, 0 < λ < 1. Then f(x0) = S(x0) = XS(x) + (1 - X)S(y) < Xf(x) + (1 - X)f(y), showing that / is convex. The strictly convex case is similar. (b) This is an immediate consequence of Theorem 4(b) and Corollary 3(a); see [Tikhomirov, p. 117]. D Remark (ix) The affine function S is called a support of f at xo; its graph is a line of support for f at x0; [RV p.12]. 4 The Cantor function is continuous, increases from 0 to 1, and is constant on the dense set of intervals [1/3,2/3],[1/9,2/9],[7/9,8/9],..., where it takes the values 1/2,1/4,3/4,...; see [CE p.187; EM2 p.13; Polya & Szego p.206]. A set, or a set of intervals, is dense if every neighbourhood of every point meets the set, or the set of intervals; [CE p.416; EM3 pp.46,434]. A function / is Lipschitz if for some M>0 and all x,y, \f(y)— f(x)\<M\y—x\\ then Μ is called a Lipschitz constant of the function; [CE p. 1091; EM5 p.532].

28 Chapter I Theorem 6 (a) f : [a, b] ι—► R is (strictly) convex if and only if there is a (strictly) increasing function д :]a, b[—► R and a c, α < с < b, such that for all α,α < χ < b, f(x) = f(c) + JXg. (b) If j" exists on]a,b[ then f is convex if and only if f" > 0; if every subinterval contains a point where f" > 0 then f is strictly convex. D See [RVpp.9-11]. D Remark (x) For applications to inequalities the conditions of (b) usually suffice; that is we are dealing with functions that are twice differentiable, with second derivative positive on a dense set6— usually the complement of a finite set; see [Bullen 1998]. It might be noted that, using the mean-value theorem of differentiation7, we can, deduce from the last property that the chord slope increases strictly; see Lemma 2. Corollary 7 (a) If r > 1 or r < 0 and f(x)=xr,x>0, then f is strictly convex; if 0 < r < 1, / is strictly concave. The exponential function is strictly convex and the logarithmic function is strictly concave. (b) If f G C2(a,6) with m = min//7,M = max/77 then both of the functions WLx tyix — f(x) and mf(x) —— are convex. Δ Δ D Simple applications of Theorem 6(b). D Example (i) Other examples are: χ logo:, χ > 0 is strictly convex; if / is (strictly) convex, twice differentiable on / and if / < 1 then 1/(1 — /) is also (strictly) convex on /. These simple examples and those in Corollary 7(a) are used in many places to prove various inequalities; [Abou-Tair & Sulaiman 1999]. Example (ii) The factorial function a:!, x > —1, is strictly convex but more is true in this case; see 4.5.2 Example (i). Example (iii) Prom Corollaries 7(a) and 3(a) we have: if χ > 0, χ ^ 1, then xr <, [>], 1 -b r(x — 1), if 0 < r < 1, [r > 1, or r < 0]. This, with a simple change of variable and notation is just 2.1 Theorem 1, that is (B), [(~B)] . Remark (xi) Corollary 7(b) has been used to extend many of the inequalities implied by convexity; [Andrica Sz Ra§a; Ra§a]. See Footnote 4 See Footnote 1.

Means and Their Inequalities 29 A simple property of convexity and concavity, that we will use later, is given in the following lemma. Lemma 8 [Kuang] If f : [0,1] н-> R is strictly increasing and either convex or concave, then for all η > 1, г=1 г=1 и D The last inequality holds since / is strictly increasing. Suppose then that / is convex and that 1 < г < η -\-1, then >/( J, since / is strictly increasing and i < η 4-1. Summing we get that So i Σ ((* -1)/(^) + (»-<+и/ф) + > > Σ K^U) - /d). This on simplification gives the first inequality. A similar argument, starting with -/( J -Ы 1 )/( ) leads ь ь n + rln + 1/ V n + 1/ Vn + 1/ to the same inequality when / is concave. D Remark (xii) This result is a special case of a slightly more general result; [Kuang; Qi 2000a]. Another useful result is the following; see [DIpp. 122-123]. Theorem 9 [Hadamard-Hermite Inequality] If f : [a, b] \-+ R is convex and if a < с < d<b, then

30 Chapter I and ;Σ/(·+;γγϊ<>->)*ϊ^Γ'· w Remark (xiii) Inequality (4) attributed Hadamard, and often called Hadamard's inequality, is actually due to Hermite8; see [Mitrinovic Sz Lackovic] for an interesting discussion of this. Recently much work has been done on generalizations of this inequality; see[PPT p. 137], [Dragomir & Pearce], [Lackovic, Maksimovic L· Vasic]. Remark (xiv) The left-hand side of (5) increases with n, having the right-hand side as its limit; see [Nanjundiah 1946]. Finally mention should be made of the important connection between convexity and the order relations of 3.3. Theorem 10 (a) [Karamata] Ifa,beIn,Ian interval in R, then Ъ^а if and only if for all functions f convex on I, 2=1 2=1 (b) [Tomic] If a,b e In, I an interval in R, then b -<w α if and only if for all functions f convex and increasing on I, 2=1 2=1 D See [MO pp.108-109]. D Corollary 11 If α and b are decreasing non-negative η-tuples such that к к 2=1 2=1 then for all ρ, ρ > 0, αρ -<w ψ'. D We can without loss in generality assume that the terms in the η-tuples are all greater than 1. Then apply Theorem 10(b)with function f{x) = epx and the η-tuples log a, log b. □ 4.2 Jensen's Inequality One of the most important convex function inequalities is Jensen's inequality; in fact it is almost no exaggeration to say that all g It is sometimes called the Hermite-Hadamard inequality

Means and Their Inequalities 31 known inequalities are particular cases of this famous result. It has been the object of much research full details of which can be found in the various references; [AIpp-10-Ц; CEp.953; DI рр.139-Щ; EM5 pp.23^-235; HLP pp.70-75; PPT pp.p-57; RVp.89]. Theorem 12 [Jensen's Inequality] If I is an interval in R on which f is convex, if η>^,2ϋΆ positive η-tuple, α an η-tuple with elements in I then: ( 1 n \ 1 n f [w~n^Wia4 ^ ^τΣ^ί**)· (J) If f is strictly convex then (J) is strict unless a is constant. D It should first be remarked that the left-hand side of (J) is well defined since the argument of / is the arithmetic mean of α with weight w_ and so its value is between the max a, and mina, in particular it is in /; see II 1.1(2). Following later usage we will refer to w_ as the η-tuple of weights. (i) This proof, by Jensen, is by induction on η and the case η = 2 is just the definition of convexity, 4.1 (1); see for instance [Hrimic 2001]. Suppose then that (J) holds for all /c, 2 < к < η — 1. <y2-f(on) + -ψ^-f ( "^ ^2wiai J , by the case η = 2, 1 n <—- 2_\wif(ai)^y ^ne induction hypothesis. The case of equality is easily considered. (w) A very simple inductive proof for the equal weight case has been given by Aczel, [Aczel 1961a,b; Pecaric 1990c]. Assume that η > 2 and that the result is known for all /c, 2 < к < п. '(ϊ;Σ>) =/(5(^ΐ0" + ^Γ=Τ)έ0, + ^ϊΣα')) by the case η = 2, \ ч г=1 ' г=1 / by the case n = 2, and the induction hypothesis.

32 Chapter I On rearranging this gives /(;Σ".)^Σ/κ>· 4 2 = 1 ' 2=1 {iii) A simple geometric induction can be given if we take as the definition of strict convexity that f" > 0 and in every sub-interval there is a point where f" is positive.; see 4.1 Theorem 6(b), Remark (x). First we must prove (J) in the case η = 2; that is, show that 4.1 (1) holds strictly ifa?^y,A^0,l. With a slight change of notation we can rewrite (J) in this case as: f(T=lx + sy) < (1 - s)f(x) + e/(y), 0 < s < 1, (J2) with equality only if χ = у. То prove (J2) consider the following function obtained from the difference between its two sides D2(x,y; s) = D2(s) = (1 - s)f{x) + sf(y) - f(T=lx + ey) where 0 < s < 1 and, without loss in generality, χ <y. Then (J2) is equivalent to D2(s) > 0, 0 < 5 < 1. Since Дг(0) = Дг(1) = 0 for some s0, 0 < s0 < 1, £>2(5ο) = 0; then 1 — sox + ^oy is a mean-value point for / on [#,y]9. Further Щ*) =/(y) - /(*) - (2/ - ^/'(T3^^ + *y) D'i{s) = -{y-x)2f"(\-=-sx + sy) By our assumption and Theorem 6(b), D is strictly concave and D'2 is strictly decreasing. Hence So is unique, with D2 positive to the left of the mean-value point, and negative to the right. Since then D2 is not constant we have that it is positive except at 5 = 0,1, which proves (J2)5 and that / is convex in the sense of 4,1 Definition 1. The case η = 3 of (J) can be written as, f(l-s-tx + ay + tz)<(l-8- t)f(x) + sf(y) + f/(s), (J3) where 0<s-H<1,0<s<1,1<£<1 with equality only if χ = у = ζ. To prove (J3) we, by analogy with the above, consider the function, D3(a?,y,*;M) = D3(a,t) = (l-8-t)f(x) + 8f(y)+tf(z)-f(l-8-tx+sy+tz) 9 See Footnote 1.

Means and Their Inequalities 33 where without loss in generality we have χ <y < z, and 0< s< 1, 0<f < 1, 0< s + ί < 1. Τ Since D3 is continuous it attains both its maximum and its minimum on Τ and if this occurs in the interior of Τ then it occurs at a turning point. Now ^D3(8,t)=f(y)-f(x)-f'(l-8-tx + 8y + tz)(y-x), ^D3(8,t)=f(z)-f(x)-f'(l-8-tX + 8y + tz)(z-x). So for a turning point at (5, t) we must have that m - fix) _ m - fix) /'(l - s-tx + sy + tz) = у — χ ζ — χ By a 4.1 Lemma 2, (f(y) - f(x))/(y - x) < (f(z) - f(x))/(z - x). So D3 has no turning points in Τ and attains its minimum on the boundary of T. However on a side of Τ the problem reduces to the previous case; for instance when t = 0, D3(x,y,z\ s,0) = D2{x,y\ s). Hence D3 attains its minimum value of zero at the corners of T, the points (0,0), (0,1), (1,0), which proves (J3). Now it is clear that this argument easily extends to the general case. D Remark (i) Of course D2(s) is defined for all 5 £ R for which (1 — s)x -\- sy e I and the argument given above shows that D'2 is strictly decreasing for all such 5. This implies that D2 is negative outside [0,1]; that is r = 0 if s = 0,1; D2{s) I >0 if 0<s< 1; I < 0 if 5 < 0 or 1 < 5. So in the case η = 2, (J2) only holds for positive weights. If η > 3 the situation is more interesting, see the Jensen-Steffensen inequality, see below 4.3 Theorem 20. More precisely if η = 2 then (~ J2) holds if either s < 0 or s > 1, see Figure 1; in the case η = 3 (~J3) will hold if either (i) 5 +1 > 1, t > 1, (ii) 5 +1 > l,i < 0, (iii) 5 +1 < 0, t < 0, or (iv) 5 +1 < 0, t > 1. Remark (ii) Proof (г), as well as other proofs, can be found in [PPT p.^S; RV p. 189], [McShane; Mitrinovic L· Mitrovic; Pop; Zajta]. Proof (iii) is in [Bullen 1994b, 1998] See also [CE p.953; DIрр.129-Ц1; ЕМ5 pp.234-235]. Remark (iii) It is immediate that (J) is equivalent to (1) and so to convexity. Remark (iv) Another form of (~J) can be found in III 2.1 Theorem 1 Proof (in).

34 Chapter I A completely different proof of the equal weight case of (J) has been given in the interesting paper by Wellstein; [Wellstein]. Given an с > 0 define Hn(c) as Hn(c) = {a; a e Rn and Σ2=ι α* = nc}> further ^ Ρ *s an го-tuple define q and q' by & = Pi/(Pi +Pi+i), 1 < i < n, g' = 1 - <?, where pn+1 = px. Using this notation and that of 3.3 Example(i) we have the following result. Theorem 13 If I is ал interval in R and if f : In Π Hn{c) н-> R is continuous at с and for alH, 1 <i <n,f(a) > f{aPi) with equality if and only if α^+ι = a* £Ьел Да) > /(се), with equality if and only if α is constant. D The proof depends on certain properties of doubly stochastic matrices. In particular, again using the notation of 3.3 Example(i) and that in Notations 7, lim Pk = - J. Details can be found in the reference. Π к—юо П Taking ρ = e and /(а) = Σ™=1 g(a>i), where g is convex this theorem gives a proof of the equal weight case of (J). As has been suggested (J) includes many of the classical inequalities as special cases, and many other inequalities are often a much disguised form of (J); see [He J K]. For instance we have the following. Corollary 14 If w_ is a positive η-tuple then 2=1 D Use the concavity of the logarithmic function. D The following interesting refinement of (J) has been given in [Vasic Sz Mijalkovic]. Let / G I, an index set, we extend the notation introduced earlier, Notations 6(xi), by defining the following function on X\ Ffe /) = F(I) = Σ ™if(ai) ~ wif ( ^Γ Σ w^ iei \ I ш Using this notation Theorem 12 can be expressed as follows. If f is convex then F is non-negative, and if f strictly convex F is positive unless α is constant. In fact more is true, as the following theorem shows.

Means and Their Inequalities 35 Theorem 15 (a) If f is convex then F is super-additive as a function of I, that jsif/1,/2Glwitii/in/i = 0, F(/i) + F(/2)<F(/iU/2). (6) If f is strictly convex then (6) is strict unless (b) If f is convex then F is super-additive as a function of w_, that is if u, v_ are η-tuples, then F(m I) + F(v; I) < F(u + υ; I) Q (a) As remarked above (J) implies that F(Ii U J2) > 0. In this inequality replace a^ by bi where bi= < —- ^Wiui, if г G /i, W7i r 11 ieh —— ^Wiui, if г G /2- Wl2 %eh Then (6) is immediate and the case of equality follows from that for (J), (b) See [Dragomir, Pecaric & Persson]. D Since (6) gives a lower bound for F(I± U /2) that is in general positive it extends (J). This is made more precise in the following corollary. In the case Ik = {1,2,... ,/c}, 1 < к < η let us write F(Jfc) = i*fc(a;t/r,/) Corollary 16 If f is convex then Fnfam f) > Fn-i&w; /)>···> F2(a; w; /) > 0. (7) In particular Fn(a;w_; f) > i^fe w; /), or more generally Fnfam f) > хтахп iwifidi) + w0f(a0) - fa + w0)f y^.^S) J ' ^ D (7) is an immediate consequence of (6) and the rest follows noting that F(In) is unchanged if the elements of a and w_ are simultaneously permuted. D Remark (v) Further extensive study of the function F can be found in [Dragomir &Goh 1997a]. Another very simple lower bound for Fn(a;w_;f) in the case of convex functions with continuous second derivatives is given in the following theorem; [Mercer A 1998].

36 Chapter I Theorem 17 If I is an interval in R on which f is convex, f e C2(I), η > 2, w_ a positive η-tuple with Wn = Ι, α an η-tuple with elements in I then for some ξ, mina < ξ < max α, Fn(&w,f) = ^„(а;ш;*2)/''(0 > °> where l2 (x) = x2. 1 n D Let а = У^ ι^α2· then, by Taylor's theorem, see 2.2 Footnote 2, for some Ρ ζί between α» and а, /(а<) = /(а) + (щ - a)f'(a) + |(а4 - а)2/"(&), 1 < г < п. Multiplying by w4 and summing over i gives Fn(a;w; f) =\ έ^ -a)2/"&) 2=1 1 n =-(^ΐϋί(α2· - a)2)/"(£),mina<£<maxa, by the continuity of/", 2=1 D Remark (vi) If in this theorem / is strictly convex then for some m > 0 we have that /" > m so that 2=1 \ n i=l / 2=1 giving an improvement for (J). Mitrinovic & Pecaric proved the following theorem; for a proof of a more general version see [PPT pp. 90-91]. Theorem 18 Let f be convex on the interval I, with А, В and r 6l, r being in an interval J such that if χ £ I then both of ((1 — X)x + Xr) A + (1 — X)(r — x)B and X(r - x)A + (Xx + (1 - X)r)B are in I. If g(x) = λ/((1 - X)x + Ar)A+(l - A)(r - x)£) + (1 - λ)/(A(r - χ)Α + (Ax + (1 - X)r)B), then g is convex and if xy > 0 and \x\ < \y\ then g{x) < g{y). Finally we give an extension of (J) due to Rooin, [Rooin 2001b].

Means and Their Inequalities 37 Theorem 19 Let I be an interval in R on which f is convex and ρ be a matrix of аШпе functions, Pij(t) = (1 — 0a*j + *Aj» 1 < *».?" < ft, 0 < ί < 1, with η η η η (Xij, Pij > 0, 1 < i,j < η, and ^a^· = ^α»5· = J^ftj = J^fti = 1> 2=1 J=l 2=1 j = l and for a given η-tuple a with elements in I we define 1 η η η 2=1 J=l Then F is convex on [0,1] and ifw_ is a positive η-tuple, t an η-tuple with elements in [0,1] j = l П 2=1 П 2=1 2=1 in particular /(^ΐ>ί)^(*)^Σ/(*).ο<ί<ι, j = l 2=1 j = l " U 2=1 Remark (vii) The last inequality is a discrete version of the Hadamard-Hermite inequality, 4.1 Theorem 9. 4.3 The Jensen-Steffensen Inequality The inequality (J) is not generalized if we allow the weights w_ to be non-negative, with of course Wn ^ 0. For if к elements of w_ are zero, 0 < к < η then (J) becomes the same result but for an (n — /c)-tuple; the case of equality when / is strictly convex has "aii elements of α that have non-zero weights are equai" instead of " α is constant"; we then will say that α is essentially constant as it is convenient to call the elements of α that have non-zero weights the essential elements of α for the given weights Also in the case η = 2 if we allow negative weights (J) does not hold, see 4.2 Remark (i). However if η > 3 real weights are possible as the following extension of (J) due to Steffensen shows; see [DIр.Ц2], [Steffensen, 1919]. Theorem 20 [Jensen-Steffensen Inequality] If I is an interval in R on which f is convex and if α is a monotonic η-tuple, η > 2, with elements in I and ifw_ is a real η-tuple satisfying Wa Wn^0, and 0 < —j- < 1, 1 < i < n, (9)

38 Chapter I then (J) holds; farther if f is strictly convex (J) is strict unless a is essentially constant. Remark (i) Condition (9) is equivalent to щ- _ w. Wn φ 0, and 0 < -?——- < 1, 1 < г < п. In addition there is no loss in generality is assuming Wn > 0 when (9) is equivalent to 0 < Wi < Wn, or to 0 < Wn - Wi < Wn, 1 < i < n. D In this proof we will assume, without loss in generality by Remark (i), that a is increasing. As some of the weights may now be zero or negative it is not as obvious as in 4.2 1 n Theorem 12 that the left-hand side of (J) is well defined. However —— Y^ гигаг· = Wn 2=1 an + У^ —-Δα2· and so by (9), and the hypothesis on a, αχ < —- У^ гигаг· < an, ~1 wn Wn ^ 2=1 1 n in particular then —— V^ гигаг· £ I- Wnt: (г) The first proof is by induction on η so let us assume the result for all /c, 3 < к < η, and assume without loss in generality that Wn > 0; this implies that Wi,Wn > 0. Further if w_ > 0 the result reduces to (J) with a value of η the number of non-zero elements in w_; so assume there is a /c, 1 < к < η, such that u>i > 0,1 < г < k, and w^ < 0. Then: г=1 \ i=i / i=k \ г=1 / г=к if( 1 vJ \ Wfc_! + ^/(afe) + -i- £ Wi/(ai). (11) WnJy~K' ' Wn 2=fc + l Now the coefficients Wk,Wk+ii · · · iwn satisfy condition (9) and so we can apply

Means and Their Inequalities 39 the induction hypothesis to the last two terms of (11) to get Now the three coefficients Wk-i/Wn, —Wk-i/Wn, 1 also satisfy (9) and the result follows by the case η = 3. It remains to consider the case η = 3. In this case we can assume without loss in generality that W3 > 0, ΐϋι,ι^ > 0,1^2 < 0 and a\ < a<i < аз; all other cases either contradict (9) or reduce to (J) in the case η = 2. Consider then (10) and (11) with к = 2; there is no need to apply (J) so they are W2CL2 + ^3 a3 equal. Then, putting а = — , we get that W3 3 1 V^ ft \ Wl (*< \ *( \\ 1 ^2/(^2) + ™зДаз) щЪ^Ы = - w^i{a2)" /(αι)) + ϊ^ Wi >-^(/(α2)-/(αι))+/(α), by(J). 1 3 Hence, putting a = —— ^ гигаг· W*u* т^ £>/Ы~/(а) > -|M/(a2) - /(а!)) + /(а) - /(a) W8 ^ ' w ~ Wj 2=1 3 w1(a2 -αϊ) W, /(a) - /(a) /(a2) - /(αχ) α·2 — οί > 0, by 4.1 Lemma 2, since а2 < а < аз, and αϊ < а < аз. This completes the proof of the case η = 3. (гг) Proof (viii) of 4.2 Theorem 12, gives a particularly simple geometrical induction of the Jensen-Steffensen inequality. While -^2(5), using the notation of that proof, is positive precisely on the interval ]0,1[, see 4.2 Remark (i), the function D3(s,£) is positive on a region larger than the interior of the triangle T. This is because D3 is continuous and positive on the whole of Τ except for the corners. Precisely D3 > 0 in the region bounded by the 0-level curve of D3 that passes through the corners of T. The region bounded by the 0-level curve depends, in

40 Chapter I general, on the values of x,y,z. Thus if χ = у it is the strip 0 < t < 1, while if у = ζ it is the strip 0 < 5 + t < 1. The question to be taken up is to find, if possible, a region larger than Τ that does not depend on x,y,z, as Τ does not so depend, and on which Ds > 0. The region we are looking for is 5 = f){(x,y,zy,x<y<z}{(s> *); Ds(x> У> ΖΊ 5> *) ^ °}· If T is a proper subset of S then Jensen's inequality will hold for certain negative values of the weights. This is just what Steffensen proved. It follows from the above that S is a subset of the parallelogram common to the two strips 0 < t < 1 and 0 < s -\-t < 1; that is the region: 0<s + t < 1, 0<t < 1. Ρ Note that this parallelogram P, unlike the triangle T, is not symmetric with respect to the variables s,t and so the condition χ < у < ζ used to prove Steffensen's extension is necessary.. Since, as we have observed, D$ reduces to a D4 on each of the sides of Τ it follows from the observations made about Di, 4.2 Remarks (i), that Ds < 0 on the extensions of these sides; that is Ds(s,t) < 0 if (i) t = 0 and 5 < 0 or 5 > 1; (ii) 5 = 0 and t < 0 or t > 1 (iii) 5 +1 = 1 and t < 0 or t > 1. In addition considerations of the partial derivatives of D3 at the corners of Τ show that Ds < 0 in the regions containing the external angles of T; that is the regions bounded by two rays on which we have just seen that D3 is negative. The tangent to the 0-level curve at the origin makes an angle Θ\ with the positive s axis where tan* - ^3^(0,0) _ (у - x) (/'(с) - /'(*)) here ^ s axis where tanΘ, - ^^ ^ Q) - ^ _ χ) (/,((/) _ fl{x)), here as below x < с < у, χ < d < ζ; as a result we have that — 1 < tan^ < 0. This implies that the line s + t = 0 crosses the 0-level curve at the origin, being on the side of T when 5 < 0. Similarly the tangent to the 0-level curve at the (0,1) makes an angle #2 with the (y _ x\ (f(c) - f(z)) positive s-axis where tan#2 = —7 ч / л// T4 л// чч,, and so — 1 < tan#2 < 0. (z-x)(f'(d)-f'(z)) This implies that the line 5 + t = 0 crosses the 0-level curve at this point, being above the curve when 5 < 0. A similar discussion at the point (1,0) leads to an angle with a tangent that is sometimes positive and sometimes negative depending on the values of ж, у, ζ, since (у - x) if (с) - f(y)) there we have, with the obvious notation that tan#3 = —-. w p. T4 . .. , (z-x)(f'(d)-f'(y)) and so as is to be expected this corner is of no interest to us. At the first two corners we have locally that Ds is positive on the sides of P. We now show that in fact Ds is positive on these two sides of P.

Means and Their Inequalities 41 Put 0(5) = D3(s, 1) when ф'(з) = f(y) - f(x) - f'(sx + sy + z)(y - x) and φη(β) = —f"(—sx -f- sy -f- z)(y — x)2. So φ is concave, zero at 5 = 0, with a unique maximum at sq < 0, where — Sqx -f- s$y -f- ζ is a mean-value point of / on [ж, у]. Now consider 7(5) = 1)3(5, —5). A similar argument shows that 7 is concave, zero at 5 = 0 with a unique maximum at s\ < 0 where χ -\- S\y — S\z is a mean-value point of / on [y, 2] So finally consider £>3(-l,l) = /(*)-/(v)+/W where h = ζ — y. Hence by the convexity of /, we get that Ds(—l,l) > 0. So Ds is positive on the sides of Ρ except at the corners of Τ and so by the general properties of Ds we have that Ds > 0 on P, being zero only at the corners of Τ ; in addition Ds attains its maximum value on one of the sides of P. In particular we have, on rewriting the inequalities that define P, that if χ < у < ζ then (J3) holds, if 0<l-s-i<l, 0< 1-i < 1, with equality only if χ = у = ζ. This is Steffensen's extension of Jensen's inequality in the case η = 3. Since, as we have seen, there is no Steffensen extension in the case η = 2, the preceding result is the first step in an inductive proof of the Steffensen theorem. As a result, to see how the induction proceeds we will consider the case η = 4. Here the function to look at is, D^(w,x,y,z)3,t,u) = DA(s,t,u) = (1-5 - t - u)f(w) + sf(x) + tf(y) + uf(z) — f (l — 5 — t — uw + sx + ty + uz) , with w < χ < у < ζ and 0<s + t + u<l, 0<t + u<l, 0<и<1. S As before the function D4 attains both its maximum and minimum values on S on the boundary of S. Unlike the case of T, and its higher dimensional analogues, the fundamental simplices, the restriction of D4 to a face of S does not immediately reduce to a case of the Steffensen inequality for a lower value of n. However as we will see, it is possible on each face to make this reduction in at most two steps. There are six cases to consider: (I) и = 0; (II) и = 1; (III) t+u = 0; (IV) t+u = 1; (V) 5+t+n = 0; (VI) s+t+u = 1.

42 Chapter I Case (I) Here a simple reduction occurs: D±(w,x,y,z\s,t,0) = D^{w,x,y\s,t), and 0 < 5 +1 < 1, 0 < t < 1. So on this face D4 > 0 by the case η = 3 . Case (II) In this case 0 < — s — t < 1, 0 < — t < 1 and we have to show that D±(w,x,y,z\s,t,l) > 0. Now, (-e-i)/W+e/(a;) + i/(y) + /^) =(-s - t)f(w) + (5 + *)/(*) - f/(я) + f/(y) + /(*), >(-5 - t)f(w) + (5 + f)/(a) + /(-te + fy + 2:), by the case η = 3, >/ (—5 — itu + s-bix + [—tx + ty + z]), by the case , η = 3, =/ (-s-ίΐϋ + 5ж + ty + 2?) , which gives this case. Case (III) Here 0 < s < 1, 0 < — t < 1 and we consider D4(w,x,y,z;s,t,—t). Now (1 - a)f(w) + β/φ + tf(y)-tf(z) > f(T=iw + еж) + f/(y) - f/(s), by (J2), >/(l — SW + sx]+ty — tz), by the case η = 3, =/(1 — 5 ги -f- 5Ж + ty — te). So D±(w, x, у, ζ; s, ί, —ί) > 0. Case (IV) Now 0 < —s < 1, 0 < ί < 1 and we must prove D±(w, x, y, z; s, t, 1 — t) > 0. Now, -8f(w) + s/(z) + f/(y)+(l - *)/(*) > - 5/H + 5Дя) + /(iy + T=f*)> by (J2), >f(—sw + sx + [ty) + 1 — ί 2]), by the case η = 3, which gives this case. Case (V) Here 0 < -5 < 1, 0 < -5 - t < 1 and /И + sf(x)+tf(y) + (-5 - t)/(2) =/И + в/(я) - e/(y) + (β + *)/(y) + (s - t)f(z), >f(w + sx — sy) + (5 + t)f(y) + (-5 - t)f(z), by the case η = 3, >f([w + sx — sy] + s -\-1 у -\—5 — tz), by the case η = 3, =/(w + sx - sy + ty + -5 -tz), which shows that D^w, x, y, z\ s, t, —s — t) > 0. Case (VI) Like Case (I) this case reduces directly to a case where η = 3 as D±{w, x, y, 2; 1 — t — it, t, it) is just £>з(ж, у, г; t, u) and 0<£ + гл<1, 0<г/<1.

Means and Their Inequalities 43 These arguments can be extended to higher values of η D Remark (ii) The above inductive proof and the geometric proof are due to Bullen, [Bullen 1970a, 1998]. The geometric proof shows that the domain of weights defined by (9) is best possible. Remark (iii) Note that as the domain defined by (9) is not invariant under permutations of the order so that the condition that a be monotone is necessary; see the comment in proof (ii). Remark (iv) A different proof can be found in [MI] ; see also [PPT p. 57], [Lovera; Magnus; Pecaric 1981b, 1984a, 1985a]. Remark (v) It was shown in the proof of the last theorem that condition (9) on 1 n the weights implies that if α in increasing then —— y^wiCLi lies in the interval [αι,αη]. The converse is also true as choosing for α the n-tuples (—1,0,... ,0), (-1,-1,0,...,0), ...,(-1,-1,. ..,-1,0), (-1,-1,... ,-1) shows. Remark (vi) For an application of the argument in proof (ii) see II 5.8 Remark (v)· 4.4 Reverse and Converse Jensen Inequalities We have seen in 4.2 Remark (i) when (~ J2) or (~ J3) holds. Various reverse inequalities have been proved for general n; we just mention two of these. Theorem 21 [Reverse Jensen] If I is an interval in R on which f is convex, if η > 2, w_ a real η-tuple, with Wn > 0, w\ > 0, Wi < 0,2 < г < η, α an η-tuple 1 n with elements in I and with а = —— \^ Wiai Ξ I then (~J) holds. If f is strictly Wn <=i convex then (~J) is strict unless a is constant. D This is a particular case of 4.2 Theorem 1; use the η-tuple (α, α2,..., an), and weights (Wn, -w2, · · ·» ~wn)- D Remark (i) This is is given in [PPTp.83], [Vasic &; Pecaric 1979a]; these authors have given similar reversals for 4.2 Theorem 15 and 4.2 Corollary 16. A special case of this result is that of the inequality between the pseudo-arithmetic and pseudo-geometric means discussed in II 5.8, and a related topic is the Aczel-Lorentz inequality, III 2.5.7.

44 Chapter I Theorem 22 [Reverse Jensen-Steffensen] Let a, /, n, be as in Theorem 21 with w_ a real η-tuple with Wn > 0. Then (~J) holds for all functions f convex on I and for every monotonic α if and only if for some m, 1 < m < n, Wk < 0,1 < к < т and Wn - Wk < 0, m < к < п. D See [PPT pp.83-84], [Pecaric 1981a, 1984b]. D The following upper bound for the right-hand side of (J) was proved in [Lah Sz Ribaric] but the proof given is in [Beesack L· Pecaric]; see also [PPTp.98], [Pecaric 1990b]. Theorem 23 If f is convex on the interval I = [a, b] and α and η-tuple with elements in I, and w_ a positive η-tuple, and ifa is as in Theorem 21 then ££щ/(в0<^/(а) + |^>). (12) 2=1 The right-hand side of (12) increases, respectively decreases, as a function of b, respectively a. In addition if f is strictly convex, there is equality in (12) if and only if α is constant. D Prom 4.1 (2) we have that /(a*) < -^/(a) + y^/(b), 1 < i < n. This clearly implies (12). The case of equality is easily deduced and the monotonicity follows by writing the right-hand side of (12) as Да) + (а - а)Щ^& or f(b) - (b - а)Щ^^-, b — α b — а and using 4.1 (3). D It should be noted that (J) and Theorem 23 can be interpreted as follows,10 see [Lob]: Theorem 24 If f is convex on the interval I = [a,b], a an η-tuple with elements in I, and w. a positive η-tuple then the centroid G of the points (aiy f(ai)) with weights Wi, 1 < г < η, (α, a), lies above the graph of f and below the chord AB where A — (a,f(a)),B = (b,f(b)); further if f is strictly convex and α is not constant these inclusions are strict. This simple geometric fact has been used by Mitrinovic & Vasic to obtain an extension of (J); see [Mitrinovic L· Vasic 1975]. Interesting comments on the history of this method of proof, the so-called centroid method, can be found in an earlier paper, [Mitrinovic Sz Vasic 1974]; see also [MPF pp.681-695], [Aczel Sz Fenyo 1948a; Beesack 1983; Beesack & Pecaric]. The centroid of the points alv..,afc is (a^—afc)/fc; the centroid of the points a1?...,afc with weights u>i,...,u>fc, where Wj>0, l<j<k, is (wi^-]—Wkak)/Wk. In particular ~а=ур— У^™ Wif(a>i)·

Means and Their Inequalities 45 Theorem 25 [Converse Jensen Inequality] If I = [a,b], and if the function f is positive, strictly convex and twice differentiable on Ι, α an η-tuple with elements in I, n > 2, and w_ a positive η-tuple then 1 n / 1 n \ (13) where if Φ = (ίΤ\μ f(b)-f(a) ^_bf(a)-af(b) b — a ' b — a then λ is the unique solution of the equation D Noting by Theorem 4.1 Theorem 4(d) that if / is convex so is (14) =f(x) Figure 2 g = λ/, λ > 0, choose λ so that the graph of g touches AB. Then AB lies below the graph of g as also does the centroid G\ see Figure 2. This gives (13), except for the determination of λ. The equation of AB is у = μχ -f ν and this line is tangent to у = g(x) = Xf(x) if Xf(x) = μχ Λ-ν and \f{x) = μ. Solving these two equations for χ leads to the equation F(x) = 0 where F(x) = μί(χ) — ί'{χ)(μχ Λ- ν). The graph of F cuts the axis in exactly one point in i" since F(a)F(b) < 0, and F is strictly decreasing. Hence there is a unique λ that is given by (14). D Remark (ii) There is equality in (13) if and only if the point of contact of у — g{x) with AB coincides with the centroid G. Now for G to lie on AB is is both necessary and sufficient that there is an S С {1,2,..., η} such that di = b,i e S,a,i = a,i £ S. If then —- 2_jWi = θ there is equality if and only if f(b6 4- a(l — 0)) = ίφ)θ+(1-θ)/(α).

46 Chapter I Using the curves у = η -f f{x) Mitrinovic & Vasic have used the centroid method to obtain the following converse inequality, [Mitrinovic &, Vasic 1975]. Theorem 26 Under the same conditions and notations as in Theorem 25 1 n / 1 n \ — £>i/(oi)<7 + /( ^гЕвдJ ' (15) where 7 = μφ{μ) + ν — f ο φ(μ). Remark (iii) Similar considerations to those in Remark (ii) can be used to obtain the cases of equality in (15). Remark (iv) A particularly simple application of this method is Docev's inequality, II 4.2 Theorem 4. Pecaric & Beesack have noted that Theorem 23 can be used to generalize Theorems 25 and 26. Theorem 27 Let J, f,a,a,2L· be as in Theorem 23, and let J be an interval containing the range of f, F : J2 1—> R increasing in the first variable, then F(wnt^fM, /<*)) <-x {^(^/(«) + H^)' Л4 (16) The right-hand side of (16) is an increasing function ofb, and a decreasing function of a. Remark (v) Theorems 25 and 26 can be obtained from Theorem 27 by taking F(x,y) — x/y, x — У respectively. The second proof of Jensen's inequality, 4.2 Theorem 12 proof (гг), gives a natural proof of a converse of (J); [Bullen 1998]. Theorem 28 [Converse Jensen Inequality] If f : [a, b] i-> R is twice differentiable with f" > 0 and if on every subinterval there is a a point where f" > 0 then .. n 1 n (17) < (1 - i0)/(a) + tof(b) - f(T^>a + t0b), where a, b are, respectively the smallest and the largest of the a^ with associated non-zero Wi, and where 1 — ίο α + t(,b is the mean-value point for f on [a, b]. There is equality in (17) only if either all the a is essentially constant or if all the w» are

Means and Their Inequalities 47 zero except those associated with α and b, and then these have weights 1 — to,t0 respectively. D From 4.1 Theorem 6(b) / is strictly convex on [a, b). For the rest of the proof we use the notations of 4.2 Theorem 12 proof (ii). Clearly the argument in that proof shows that D2(s) attains a unique maximum at s = so, so that D2(s) < D2(sQ) with equality only if s = sq. So if 0 < s < 1, 0 < (1 - s)f{x) + sf{y) - f(T^s~x + sy) < (1 - s0)f(x) + s0f(y) - f(l-s0x + s0y) with equality on the left, (J2), only if χ = у , or on the right, a converse of (J2), if χ = у or if s = s0. Since D2{s) = f(y) - f{x) — (y — x)f'(l — sx + sy) and Дг^о) = 0 we see that 1 — so x 4- so2/ is ^ne mean-value point of / on [a, b]. This gives (17) in the case η = 2. Similarly the argument in the same proof shows that Ds(s) attains its maximum value on the boundary of T, and not at the corners where it attains its minimum value, 0. As pointed out in that proof on each edge of Τ the problem reduces to a case of η = 2. So from the above D3 has a local maximum on each edges and the question is, which is the largest of the three? If 0 < s < 1 and we consider^ as a function of χ then D'2{x) = (1 — s) (f(x) — /'(1 — sx -f sy)) , which is negative; while if we consider D2 as a function of у then D2(y) = s {f'(y) — /'(1 — sx 4- sy)) , which is positive. Hence if χ' < χ < у < у' with not both χ' = χ and у' = у then D2(x'}y';s) > D2(x,y;s), 0 < s < 1; in particular the maximum value of D2{x'\ y'\ s) is larger than that of D2(x, y; s). Hence the maximum of D3 occurs at (0, ίο) where 1 — to χ + t0z is the mean-value point for /on [ж, ζ], since we have taken χ = min{x, ?y, 0}, ζ = тах{ж, у, ζ}; that isifO<s<l,l<£<l,0<s + £<l then (1 - s - t)f(x) 4- sf{y) 4- */(*) - f(l-s-tx + sy + tz) <(1 - *о)/(*) + ίο/W - /(Τ^ϊο"χ + ίο*), with equality only \ix = у = z, or s = Q,t = tQ. This gives (17) in the case η = 3. Clearly the method extends readily to general n. D Remark (vi) (J), as well as a converse of Giaccardi, [Giaccardi 1955], have been used by several authors to obtain bounds for various functions; see for instance [Afuwape L· Imoru; Keckic L· Vasic). Remark (vii) For a different kind of converse to (J) due to Slater see [PPT pp. 63}; [Slater].

48 Chapter I Remark (viii) Inequality (J) and its converse have been objects of much recent study. Interesting generalizations can be found in [DI pp.139-142; MPF pp. 1-20; PPT pp.43-106], [Pecaric], where there are further references and many details, and in [Abou-Tair Sz Sulaiman 1998; Alzer 1991g; Barlow, Marshall Sz Proschan; Dragomir 1989 a,b, 1991, 1992a, 1994a, 1995b; Dragomir L· Goh 1996; Dragomir Sz Ionescu 1994; Dragomir Sz Milosevic 1992, 1994; Dragomir L· Toader; Dragomirescu 1989, 1992a,b, 1994; Dragomirescu Sz Constantin 1990, 1991; Fu, Bao, Zhang Ζ L· Zhang Y; Gatti 1956a; Imoru 1974a, 1975; Lin Y; Marshall L· Proschan; Mitrinovic & Pecaric 1987; Mond Sz Pecaric 1993; Needham; Nishi; Pecaric 1979, 1987c, 1990a, 1991a,b; Pecaric Sz Andrica; Pecaric L· Beesack 1986, 1987; Pecaric L· Dragomir 1989; Petrovic; Pittenger 1990; Rusmov; Schaumberger L· Kabak 1989,1991; Sirotkina; Takahasi, Tsukuda, Tanahashi Sz Ogiwara; Vasic Sz Pecaric 1979c; Vasic L· Stankovic 1973, 1976; Vincze I]. 4.5 Other Forms of Convexity 4.5.1 Mid-point Convexity Definition 29 If I is an interval in R then f : I \—> Ш is said to be mid-point convex, or J-convex, on i" if for all x,y e. I, Remark (i) Since (18) is easily seen to imply (1) for a dense subset of A it is immediate that a continuous mid-point convex function is convex; the converse is false, see [RV p.216]. However very mild restrictions on a mid-point convex function do imply its convexity; see [RV p. 215].11 Theorem 30 If f is mid-point convex on I and bounded at one point of I then it is convex. 4.5.2 Log-convexity Definition 31 If I is an interval in Ш, then a function f : I \-+ R+ is said to be log-convex or multiplicatively convex on / if logo/ is convex, or, equivalently if for all x,y e I and all λ,0 < λ < 1, f(Xx + T^~Xy)<fx(x)f1-x(y). Example (i) The function x\,x > —1, is strictly log-con vex; see [Artin]. See Notations 9.

Means and Their Inequalities 49 Lemma 32 (a) A log-convex function is convex (b) If f > 0 then f is log-convex if and only if g(x) = eaxf(x) is convex for all ael. D (a) This is an immediate consequence of 4.1 Theorem 4(f). (h) See [Alp.19]. D Example (ii) The converse of (a) is false as f{x) = ж3/2, χ > 0, shows. Remark (i) See also [DIp.163; RVpp. 18-19], [Artin рр.7-Ц]. 4.5.3 A Function Convex with respect to Another Function Definition 33 if/ С I is an interval and f,g : J ι-» R are continuous, and g strictly monotonic, J = g[I], also an interval, then f is said to be (strictly) convex with respect to g if any of the following equivalent statements hold: (a) f ° 9~X ls (strictly) convex; (b) for some function к (strictly) convex on J, f = к о д; (c) the curve χ = g(t), у = f(t), t £ I, is (strictly) convex. Remark (i) Note that the curve in (c) is actually a graph; see 4.2 Remark (ii). Remark (ii) This idea is a convenient way of expressing a property that occurs naturally; see [HLP p.75], [Cargo 1965; Mikusinski]. Example (i) In this terminology / is log-convex if and only if log is convex with respect to /_1; or in the terminology of the 4.5.2 Definition 31 we could say / is convex with respect to д as д~г is /-convex. Example (ii) / is convex if and only if it is convex with respect to a non-constant affine function; see 4.1 Remark (viii). Example (iii) / is convex with respect to д on I if and only if for all Xi, 1 < г < 3, three points of I with x\ < x<i < xs, 1 д(хг) /(χι) 1 g(x2) f{x2) 1 д(хз) Джз) >0. This is a natural extension of 4.1 (2**). Although the conditions for / to be convex with respect to g follow from conditions for convexity the following condition has been obtained by Mikusinski, and in a different way by Cargo.

50 Chapter I Theorem 34 If g e C2(J), and f e C2(I), where I, J are as in Definition 33, and if fid' are never zero then a sufficient condition for f to be convex with respect to g is that 9' - /' ' Remark (iii) A geometric interpretation of this property is given in [Mikusinski]. If / is convex with respect to g on I = [a, b], g strictly increasing, and f{a) = g(a), /(&) = g(b) then f{x) < g{x), a < χ < b\ see IV 2 Corollary 8. This extends the property of convex functions in 4.1 Remark (iii); see also [RV pp.240-246]. Example (iv) It is easy to check that log ж is convex with respect to 1 — х~г and using the previous remark this implies the left-hand side of 2.2 (9). 4.6 Convex Functions of Several Variables To extend the concept of convex function to higher dimensions we need to make the nature of its domain more precise. Definition 35 (a) A set U С Rn, η > 1, is said to be convex if ae U and beU implies that (1 - \)a + XbeU for all λ, 0 < λ < 1. (b) If U С Rn, η > 1, then the convex hull of A is the smallest convex set that contains A. Example (i) If U is a finite set U = {al5... afe}, say, then the convex hull is the set {χ; χ = Σί==1 Щ^, w_ > 0, Wk = 1}, the set of all convex combinations of the points of U. Example (ii) In Μ a set is convex if and only if it is an interval. Example (iii) If / : i" i—* Ж then / is convex if and only if the set Ε С R2, Ε = {(χ, у); х е I, у > f(x)} is convex. This set is called the epigraph of f, and the definition easily extends to real-valued functions defined on subsets of Mn, n> 2. By replacing the interval i" in R by a convex set U in Mn, η > 2, and the points x,y οϊ I by points u, y_ of U we can easily extend 4.1 Definition 1 to functions / : U -► R, and 4.5.2 Definition 31 to functions / : U -> R^; [DIp.63]. Jensen's inequality, 4.2 Theorem 12, and Theorem 18 extend to this situation with the same proofs, and 4.1 Theorem 4(a), (b) is valid in the following form; see [RV pp.93, 116-117]', and 4.5.1 Theorem 30 has a natural extension.

Means and Their Inequalities 51 Theorem 36 If f : U i—> R is convex on the open convex set U in Rn, η > 2, then f is Lipschitz on every compact subset ofU and has urst order partial derivatives almost everywhere that are continuous on the sets where they exist. There is no single criterion for investigating the convexity of functions of several variables. It is therefore useful to define some ideas that will help in special situations, and to list several criteria for convexity. Definition 37 (a) A set U in Rn is called a cone if и е U implies that for all λ > 0, we have Xu e U. (b) A function f denned on a cone is homogeneous of degree α if for all λ > 0 we have f(Xu) = \af(u). (c) If f is twice differentiable the matrix Η = (f"j)i<i,j<n is called the Hessian matrix of f. (d) The directional derivative of / at u0 in the direction ν is when the limit exists. Remark (i) A function / that is homogeneous of degree 1 is just said to be homogeneous. Remark (ii) If / is homogeneous of degree α and is differentiable then the partial derivatives of / are homogeneous of degree а — 1 and we have the Euler's theorem on homogeneous functions : uif[(u) 4- ■ · ■ 4- unf'n{u) = af(u). Example (iv) The sets R™, (R^j_)n are cones. Example (v) If / is differentiable at г^ then f+{Uo;v) = V/(2Xo).v. Theorem 38 A homogeneous function is convex on the cone U if and only if for all u,v e U, f(u + v)<f(u) + f(v), (19) and is strictly convex if and only if (19) holds strictly for all u,v e U,u^v_. Theorem 39 If f is convex on the open convex set U then f+(u0',u — u0) exists and /Ы>/Ы + ДК;м-м0)· (20)

52 Chapter I If f is strictly convex and homogeneous then there is equality in (20) if and only if и ~ Uq. Conversely if f is differentiable on U and (20) holds, strictly, for all u0, и е U then f is convex, strictly convex. Theorem 40 (a) A function is convex if and only if its epigraph is a convex set. (b) If f and д are convex, strictly convex, then so is af 4- β 9 for all α, β > 0. (c) If fa, a e A, is a family of functions convex on U then f = suPq,^^ is convex on {u; u^U, f(u) < oo} (d) If f is convex and monotone in each variable on the parallelepiped rn<u< M, and if each function qi, 1 < i < n, is convex if f increasing for the ith variable, and concave if f is decreasing for the ith variable, and ifrrii<gi<Mi, 1 < г < η, then /(<7i,... gn) is convex. (e) If f is convex, strictly convex, and homogeneous of degree α, α > 1, and positive except possibly at the origin then f1^ is convex, strictly convex, and homogeneous. (f) Suppose that f is a homogeneous function on a cone in the set {a; an > 0} and convex as a function of {u\,..., nn_i, 1) then f is convex. (g) Let f Ε C2(U), U an open convex set in M.n,n > 2. A function f is convex on U if and only if the Hessian matrix Η of f is negative semi-definite; if Η is positive definite on U then f is strictly convex. Proofs of the Theorems 38-40 can be found in the standard references; [RV pp. 94-95, 98, 103, 117], [Rockafellar pp.23-27, 30, 32-^0, 213-2Ц], [Soloviov]. Remark (iii) It is easy to state an analogous result for concave, strictly concave, functions. ( f" f" λ Remark (iv) If η = 2 then Η = ( },} }2 ι and it is non-negative definite if V /12 /22 / and only if /i'i > 0, [or/& > 0], and /ίΊ/& - /12 > 0; (21) Η is positive definite if the inequalities (21) are strict; see [HLP pp.80-81). The following examples are important for later applications. Example (vi) Let w. £ (MiJ_)n and ρ > 1 then φ(α) = Σ™=1 що^ is strictly convex on (R^_)n. This follows from the strict convexity of f{x) = xp, χ > 0, ρ > 1, and Theorem 40 (b). Example (vii) If ш and ρ are as in then previous example then 7(a) = φ1^ρ(α) = convex on (IRl)n. This follows from Example (vi) and Theorem 40 (e) since φ is homogeneous of degree p..

Means and Their Inequalities 53 Example (viii) If w_ is as in then previous examples then χ (α) = ΠΓ=ι ai is concave on (M?J_)n, strictly if η > 1. The concavity is trivial if η = 1 so suppose that we have concavity for η — 1 and write П λ .W-n-i/W-n х(а) = ^«(П«ГЛ-») -" ". (22) i=l Now g{x) = x(Wn/Wn\x > 0 is strictly concave, by the induction hypothesis ΠΓ=ι ai " ^s concave5 and Мж) — xWn-l/,Wn is strictly concave. By Theorem 40 (d) the second term on the right-hand side of (22) is strictly concave. So using Theorem 40(f) χ is strictly concave on A; see also [Rockafellar pp. 27-28]. Example (ix) A further example of a concave function is p{g}^rn), where ρ is a polynomial that is homogeneous of degree m; see III 2.1 Remark (iii). Inequality (20) can be extended in the case of homogeneous convex functions to the following inequality that is called the support inequality] [Soloviov]. Theorem 41 [Support Inequality] If f is a convex homogeneous function of the cone U then for all u,v_e U, f(u) > f'+(v;u); (23) if f is strictly convex then there is equality in (23) if and only if и ~ v_. D First note that f(y_-l· \(u — v_)) >f(v 4- Xu) — f(Xv), by convexity, =f{l + Xu) — Xfill), by homogeneity. So w, ч г f(v + X(u-v))-f(yd /+teu-20=Ahm+ , > Ит f(v + Xu)-Xf(v)-f(vd ~ λ^0+ λ =f+(v;u)-f(vd- Then (23) follows from (20). The case of equality follows from that in Theorem 39. D Remark (v) If / is concave then (~23) holds. Further if / is differentiable then (23) is just f(u) > V/(u).u. (24)

54 Chapter I 4.7 Higher Order Convexity Let αΐ}0 < г < η, η > 0, be (n -f 1) distinct points from the real interval i" then if / : i" \—> R the n-tii divided difference of f at these points is 12 [a; /] = [a; /]n = [a0,..., an; /] = V * ~f ш №) where w(x) = Ъ7п(я;а) = Щ=о(ж ~ a*)· Remark (i) It is worth noting that //(αο) /(αϊ) det [s;/]n = V ι / «ο «Ϊ / „n—1 „n—1 det V 1 see for instance [Popoviciu 1934a]. Example (i) b;/]i Ь;/Ь = /(αο) + /(αι) ~ Дао) αϊ - αο /(«ι) «г1 i J «η „n-1 + · /(«2) (25) (α0 - αι)(α0 - α2) (αϊ - α2)(αχ - α0) (α2 - α0)(α2 - αϊ) Remark (ii) The last expression shows that the left-hand side of 4.1(2*) is just [жьж2,ж3;/]2. Example (ii) If f{x) = xp,p = —1, 0,1, 2,... then te/]n = { ( 0, ifp = 0, l...,n- 1; 1, ifp = n; ^ (a^a*1 ...akn"), if ρ = n +l,n + 2,...; feoH Η^τι=ρ—™ (-1)" i, αοαι ... an See [Milne-Thomson pp.7-8]. Ίϊρ= -1. 12 Note that in this context a, is an (n + l)-tuple.

Means and Their Inequalities 55 Lemma 42 (a) [α0,αι,α2;/ρ]2 = ρ(α0)[α0,αι,α2;/]2 4- [αϊ, α2;/]ι[α0, αϊ; c/]i 4- /(α3)[α0, αϊ, аз; р]2 (bj ifn> 1 then ^^[α;/]η = |ώ;/]η-ι - fe£;/]„-i (26) D Simple calculations prove these identities. D Remark (iii) If / and g have second order derivatives then on taking limits the identity in (a) reduces to (fg)" = gf + 2/V 4- fg". The definition of [dlq ,..., an; /] can be extended to allow for certain of the points to coalesce, the so-called confluent divided difference, provided we assume sufficient differentiability properties for / so as to allow the various limits to exist. If щ 4-1 of the elements of the η-tuple α are equal to a^, 0 < г < га, then n0!...nm!anoa0...^n-amL Example (iii) For instance it can easily be checked that ΐα,α,Ο,Ο,/j- (ь_а)2 See [Milne-Thomson pp.12-19]; [Horwitz 1995]. Definition 43 If I is a real interval then f : I \-> Ш is said to be η-convex on i", η > 0, if for all choices of (n 4-1) distinct points from I. [α;/]η>0. (27) If instead (~27) holds we say that f is η-concave on i". Further if (27), respectively (~27)j is always strict we say that f is strictly η-convex, respectively strictly n- concave. Remark (iv) If η = 2 then, by Remark (ii), Definition 42 is equivalent to 4.1 Definition 1 so 2-convex functions are just convex functions. Also from Example (i) 1-convex functions are just increasing functions; and, since [a;/]o = /(ao), 0-convex functions are just non-negative functions.

56 Chapter I Theorem 44 (a) A function is both η-convex and η-concave if and only if it is a polynomial of degree at most (n — 1). (b) A function f is η-convex, η > 2 if and only if /(n~2) exists and is convex. D A proof of this theorem together with many other details can be found in [Aumann & Haupt pp.271-291]. □ Remark (v) In particular if / is η-convex, η > 2, and if 0 < к < η - 2 then /W is (n — /c)-convex; f±~ exist, are increasing and /_ ~ < /+~ . Lemma 45 (a) If f^ > 0 on the interval I then f is η-convex on I, and if every subinterval of I contains a point where f^ > 0 then f if strictly η-convex on I. (b) If f is η-convex, η > 2, and ifa,b are η-tuples with distinct elements and such that a<b then [α;/]η-ι < [kf]n-i\ when f is strictly η-convex this inequality is strict. (c) If f is η-convex, η > 2, it is strictly η-convex unless on some sub-interval f is a polynomial of degree at most (n — 1). (d) If f is a polynomial of degree at least η and if f^ > 0, then f is strictly n-convex. (e) If for all h ^ 0 small enough and all χ, α < χ < b, we have for a bounded function f that [x,x + h,...,x + nh-J]n = ~ έ(-1)"""' (") f(x + ih) Ζ Ο, then f is η-convex on ]a,b[. If the inequality is always strict then f is strictly η-convex on ]a, b[. D (a) This follows from Theorem 44(b) and the case η = 2, 4.1 Theorem 6(b); [Bullen 1971a]. (b) This is a consequence of (26); see [Bullen 1971a]. (c) and (d) follow from (b) and Theorem 44(a); [Bullen Sz Mukhopadhyay p.314]. (e) Since / is bounded then it is continuous, [RV p.239], and for the rest see [Popoviciu pp.48-49]. D Example (iv) Using Lemma 45(a) we easily see that ex is strictly η-convex for all n, and log ж is strictly η-convex if η is odd, and strictly η-concave if η is even, Example (v) Similarly xr, χ > 0, is strictly η-convex if sign (ΠΓ=Γο (r ~ *)) ~ 1· In particular this function is 3-convex ifr>2or0<r<l, and is strictly 3-concave if 1 < r < 2 or r < 0.

Means and Their Inequalities 57 Remark (vi) The property in (b) generalizes 4.1 (3). Remark (vii) The property in (e) generalizes J-convexity, 4.5.1, and a function with this property is said to be η-convex (J), or weakly n-convex. Part (e) extends 4.5.1 Theorem 30. Remark (viii) An extensive study of higher order convexity can be found in [Popoviciu]; see also [DI pp.190-191; PPT pp.15-17; RV pp.237-240]. 4.8 Schur Convexity Definition 46 A function f : In ь-» R, where I is an open interval in R+, is said to be Schur convex on In if for all a,beln, έ-!α=>/(6)</(α); (28) if the inequality is always strict when b -< α and α is not a permutation ofb then f is said to be strictly Schur convex on In. Remark (i) If — / is Schur convex, strictly Schur convex, on In then / is said to be Schur concave, strictly Schur concave on In. Remark (ii) In proving Schur convexity we can often assume that η = 2. This follows from 3.3 Lemma 13; see [MO p.58]. Lemma 47 (a) If f is Schur convex on In then it is symmetric. (b) If fi, 1 < г < m, are all Schur convex on In and h : Rm i-> R is increasing in each variable then h(fi,..., fm) is Schur convex on Iй. (c) In particular with the assumptions in (b) min{/i,..., /m}, max{/i,..., fm}, and Πΐϋι fi are Schur convex on In. (d) If f is Schur convex and increasing, in each variable, on In and if д :1и I is convex then /(ρ(αι),..., д(ап)) is f is Schur convex on In D All the proofs are straightforward; see [MO pp.60-63] D Theorem 48 [Schur-Ostrowski] If f : In ь-» R, where I is an open interval in R+, has continuous derivatives and is symmetric then f is Schur convex on In if and only if for all г Ф j (α. - aj)(fl(o) - ή (a)) > 0 for all а в Г Π See [MO pp.56-57]. Π

58 Chapter I Theorem 49 If f : In i-> R, where I is an open interval in R+, is symmetric and convex then it is Schur convex. D By Remark (ii) we can assume that η = 2 when for some λ, 0 < λ < 1 b\ = (1 — λ)αι + λα2, Ь2 = λα ι + (1 — λ)α2; see for instance 3.3 Lemma 13. So /GO =/((1 ~ λΚ + λβ2, λ^ + (1 - λ)α2) =/((1-λ)(α1,α2) + λ(α2,α1)) < (1 - λ)/(αι,α2) 4- λ/(α2,αι), by convexity. =/(α), ЬУ symmetry. Π There is an extensive literature on this subject; see in particular [A I pp. 167-168; BB ρ 32; DI pp.228-229; EM6 p.75; MO pp.5^-82; PPT pp.332-336). 4.9 Matrix Convexity If I is a interval in R then a function / : i" \-+ R is said to be a increasing matrix function of order η if for all Α, Β Ε Τί^ with eigenvalues in J and A < В we have that f(A) < f{B)13. Remark (i) The usages decreasing matrix function and monotone function are easily defined and if a function is monotone for all orders it is said to be operator monotone. Example (i) If f(x) = xp, 0 < ρ < 1 or g(x) = log χ then / and g are operator increasing on R; see [DI p.182; MO pp.463-467; RV pp.259-262). Remark (ii) The above definition can be made without the restriction to positive definite matrices but the examples given above need positive semi-definiteness for /, and positive definiteness for g. If J is a interval in R then a function / : I i-> R is said to be a a convex matrix function, of order n, if for all А, В e H^ with eigenvalues in I and all λ, 0 < λ < 1, /(Γ^λ A + XB) < T=Xf(A) + Xf(B). (29) Remark (iii) If the opposite inequality holds then the function is said to be a concave matrix function, of order n. A function that is convex, concave, of all orders is said to be operator convex, concave] [DI p.64; MO pp.467-474; RV pp.259-262). The various matrix concepts are defined in Notations 7

Means and Their Inequalities 59 Example (ii) If f(x) = x2,x \ 1/y/x then / is operator convex ; if f(x) = y/x then / is operator concave. Remark (iv) The comments in Remark (ii) apply here as well. The topic of operator monotone and operator convex functions has a large literature; in addition to the above references see [Furuta], [Ando 1979; Kubo L· Ando).

II THE ARITHMETIC, GEOMETRIC AND HARMONIC MEANS This chapter is devoted to the properties and inequalities of the classical arithmetic, geometric and harmonic means. In particular the basic inequality between these means, the Geometric Mean- Arithmetic Mean Inequality, is discussed at length with many proofs being given Various refinements of this basis inequality are then considered; in particular the Rado-Popoviciu type inequalities and the Nanjundiah inequalities. Converse inequalities are discussed as well as Cebisev's inequality Some simple properties of the logarithmic and identric means are obtained 1 Definitions and Simple Properties 1.1 The Arithmetic Mean Definition 1 If a = (αϊ,..., an) is a positive η-tuple then the arithmetic mean of a is oi ( ч αϊ Η f-an %п(Ю = . (1) η This mean1 is the simplest mean and by far the most common; in fact for a non- mathematician this is probably the only concept for averaging a set of numbers. The arithmetic mean of two numbers a and b, (a -f b)/2, was known and used by the Babylonians in 7000 B.C., [Wassell], and occurs in several contexts in the works of the Pythagorean school, sixth-fifth century B.C. For instance in the idea of the arithmetic proportion2 of two numbers, a, b, 0 < a < b, the number χ such that χ — α : b — χ :: 1 : 1, when of course χ = (α -f b)/2.3 Aristotle, also in the sixth century B.C., used the arithmetic mean but did not give it this name. Another interpretation arises from the picturing of addition as the abutting of two line segments. One then asks what line segment when abutted to itself will More precisely called the arithmetic mean with equal weights; see Remark(x) below 2 The notion of a proportion, between four positive numbers, lengths, areas etc. is somewhat archaic but has a long history, see[Euclid Book V, Heath vol 1.pp.84.-90, 384-391] If A,B,C,D are positive numbers then A:B::C:D, read A is to В as С is to D, is equivalent to A/B=C/D In this form the arithmetic mean is one of the ten neo-Pythagorean means; see VI 2.1 4 and 1 2 below. 60

Means and Their Inequalities 61 produce the same length as the abutting of two given line segments.The idea of arithmetic mean is also found in the concept of centroid used by Heron, and earlier by Archimedes in the third century B.C; [Heath vol.IIp.350]. In the notation introduced in Definition 1 the η-tuple may be replaced by some specific formulation such as 2ln(ai,..., αη), or 2ln(oi, 1 < i < n). Further if η = 2 the suffix is omitted unless it is needed for clarity4; thus we will write 21 (a; w) etc., when η = 2. When there is no ambiguity either a, or the subscript n, or both may be omitted. Convention 1 If α is an η-tuple, η > 2, and if 1 < m < n, we will write, whenever there is no ambiguity, Convention 2 The various notations introduced above will apply in various contexts throughout this work. Clearly (1) does not depend on a being positive, and many of the properties of 21 can be deduced without this assumption, see below Remarks (v), (x), 1.3.3, 1.3.4, 1.3.9 and 2.4.5 Remark (v). In fact the whole discussion can take place in a very general context, see for instance VI 5 and [Anderson L· Trapp]. However we have the following convention that will hold throughout the rest of this book unless otherwise indicated. Convention 3 All n-tuples and sequences will be positive unless specifically stated otherwise; that is if α is an n- tuple then unless otherwise stated, α e(R+)n. The question of more general η-tuples will be discussed in various places later; for a situation where the elements of α are complex see [Majo Torrent}.5 If the elements α are in Q then 2in(a) e Q, and because the above definition is so elementary there is a certain point in using only the simplest tools to deduce properties of this mean. Of course non-elementary proofs besides having an interest in themselves suggest methods of generalization, and are often simpler than the more elementary approaches. Some elementary properties of 21 are listed in the following theorem; they are all obtained by easy applications of simple properties of positive real numbers. There are many means that are only defined when n=2, see VI 2, and dropping the suffix makes comparisons easier to read. However the usage with n=2, or even n=l, is useful from time to time in inductive arguments. Convention 3 will not apply in Chapter IV.

62 Chapter II Theorem 2 If h is a real η-tuple, α,α + h,b η-tuples, and if λ > 0 then: (Ad) [Additivity] 2ln(a + b) = 2ln(a) + 2ln(6); (As)m [id-Associativity] 2in(a) = 2tn(2tm,. ..2tm,am+i,... ,an), 1 < m < n; (Όο,) [Continuity] lim2ln(a + ft) =2tn(a); (Ήο) [Homogeneity] 2in(Aa) = A2im(a), λ > 0; (7л J [Internality] mina < 2ln(a) < max a, (2) with equality if and only if α is constant; (Mo) [Monotonicity] a<b => 2tn(a) <2tn(b), with equality if and only if а = Ь; fReJ [Reflexivity] If a is constant, (ц = a, 1 < г < η, tiien 2tn(a) = a; fSyJ [Symmetry] 2tn(ai,.. . ,an) is not changed if the elements of α are permuted . Remark (i) When m-associativity holds for all ra, 1 < m < n, as it does for the arithmetic mean, we call it the property of substitution. Remark (ii) Clearly it is inequality (2) that justifies the name of mean, see [B2 p. 230). While (2) will be called the property of internality the full property given in (In) will be called strict internality. Remark (iii) The word monotonic used in (Mo) is sometimes replaced by monotone, or by isotone, see [B2 p.230). The concept can also be considered in a strict form, strictly monotonic, etc. Remark (iv) The properties listed in Theorem 2 are not independent. For instance (In) is implied by (Mo) and (Re). Remark (v) It is useful to note that these properties of the arithmetic means hold if the гг-tuple α is allowed to be real. Almost all the means we will discuss in this book satisfy some or all of these conditions. The properties (Co), (Re) and (In) are so basic that we would consider

Means and Their Inequalities 63 them essential in any possible definition of a mean; the property (Ho) is a very common property but does not always hold; for examples see IV. Others, such as (Ad) are in some sense characteristic of the arithmetic mean; see VI 6. Remark (vi) While (Ad) gives a simple relation between 2ln (a), 2ln (b) and 2ln (a+ b) it is much more difficult to obtain one between 2ln (a), 2ln (b) and 21η(αέ)> but one is given later, see 5.3, and is known as Cebisev's inequality. A very natural extension of Definition 1 is suggested when some of the elements of α occur more than once or, from a practical point of view, if some are considered more important than others. Definition 3 Given two η-tuples a, w_, the weighted arithmetic mean of a with weight, or weights, w_ is wiaH \-wnan 1 ^A «»te^ = Wl + ... + Wn = Wnywiai' (3) It is easily checked that this more general mean has all the properties listed in Theorem 2 except (Sy). Instead this more general mean has the following property: ( Sy*) [Almost Symmetry] 2infe W.) is not changed if the α and w_ are permuted simultaneously. Remark (vii) The name arithmetic mean will normally refer to (3), and if we wish to specify (1) it will be referred to as the arithmetic mean with equal weights. Remark (viii) Using this notation Jensen's inequality, I 4.2 (J), can be written as: /(Оп(а;ш)<Оп(/(а);ш). (4) A refinement and strengthening of inequality in (2) is given in the following result Lemma 4 (a) Ifa,b and w are η-tuples, then min (ab-i) <^n(/f-) <шах(аГ1). (b) IfWn = 1 and η > 2, then max a — У1п(а', w_) > — у, \уаг ~ y/aj) · l<i<j<n D (a) Let m = min (a b_1) and Μ = max (a b-1), then m < -^ < M, 1 < г < n, or mwibi < diWi < Mwibi, 1 < г < η; summing over i gives the result.

64 Chapter II (b) Let S(a) denote the difference between the left-hand side and the right-hand side of the inequality in (b); we have to show that S(a) > 0. Put w — minw, and assume without loss in generality that a is decreasing. Now, η /_] (y/αϊ — y/uj) ={n - l)2_\ai— ^ /_^ y/didj l<i<j<n i=i l<i<j<n η <(n — 1) ^α^ — 2 X. min(ai,aj) i=i l<i<j<n η = Y^{n + l-2i)ai. η + 1 — 1% So S(a) = ai - 2infe W.'), where w[ — wi -\ —w, 1 < i < n, and W'n = 1. Hence by (2) S(a) > 0 as was to be shown. D Remark (ix) Result (a) is due to Cauchy, see [AI p.204), and analogous inequalities can be given for the geometric and harmonic means defined in the next section. Remark (x) Part (b) is due to Alzer and can be extended to real η-tuples a; [Alzer 1997d]. Remark (xi) Another generalization is given in [Bromwich pp.242, 4^^-420, 473- 474], [Bromwich]. Remark (xii) For an amusing discussion of weighted arithmetic means see [Falk &Ваг-НПЫ}. 1.2 The Geometric and Harmonic Means Two other means of a very elementary nature have been in use for a long time. Like the arithmetic mean they arise naturally in many simple algebraic and geometric problems, some of which are to be found in Euclid and in the work of the Pythagorean school. Thus in that 2ab a + b school's theory of harmony study is made of the proportions a: -:: —-—: b; fur- a + b 2 ther in Euclid we have the study of the proportions x — a: b—x:: a: x when χ = Vab, the geometric mean of a and b, and χ — a : b — χ :: a : b when χ = 2ab/(a + b), the harmonic mean; see also 1.1 Footnote 3. The name geometric is probably based on the Greek picturing of multiplication of two numbers as the area of a rectangle. One then asks what number multiplied by itself will produce a square of the same area as the rectangle obtained by multiplying two given numbers; [Aumann 1935b].

Means and Their Inequalities 65 Definition 5 Given two η-tuples a, w_, the weighted geometric mean, respectively harmonic mean, of α with weight, or weights, w is 0n(«;m) = (n«r)1/W". (5) i=i respectively fin (a; w) = n n . (6) Remark (i) The other variations of notation introduced for the arithmetic mean will be used here, and with other means introduced later. Thus if w_ is constant we get the means with equal weights, written Φη(α),ί)η(α); and note should be made of 1.1 Conventions 1, 2 and 3. Remark (ii) The replacement of the arithmetic means in inequality 1(4) by other means has been used by some authors to define another form of convexity; thus we would say that a function / : R+ ι—> R is convex relative to the geometric mean, or just geometrically convex, if for all η-tuples α and w; f(®n{a;w)) <®n(f(a)-w). A function / is geometrically convex if and only if / о ехр is log-convex; [Jarczyk L· Matkowski; Kominek L· Zgraja; Matkowski L· Ratz 1995a,b; Niculescu; Thielman; Toader 1991b}; see also III 6.3 . Remark (iii) The harmonic mean of two numbers with equal weights was originally named the sub-contrary mean; it was renamed by Archytas and Hippasus; see [Eves 1976; Heath vol.Ip.85}6. The following simple identities should be noted. Ыа;ш) = («„ (α"1; Μ))""1 or ^ ^^1; (7) г=1 ®n(g>]w) = exp (&n(bga;;u;)J, or log (<5п(а;ш)) = ^n(loga;u>). (8) Because of the particular simplicity of (7) the properties of the harmonic mean will often not be investigated in detail. In any case all of these means turn up again later as special cases of the more general power means; see III 1. Remark (iv) For other relations between the arithmetic and geometric means with some geometric interpretations see [Jecklin 1962a; Usai 1940a}. The frequency of the octaves being in the ratios 1,2,3... the harmonic mean of 1 and 2, namely 4/3, gives the frequency of the fourth, that of 1 and 3, namely 3/2, gives the frequency the fifth etc. Further 8, the number of angles in a cube, is the harmonic mean of the number of edges, 12, and the number of faces, 6. All good reasons for the name change; see [Heath vol.1 pp.75- 76, 85-86], [Wassell].

66 Chapter II Theorem 6 The geometric and harmonic means have the properties of (Co), (Ho), (Mo),(Re), (Sy*)} (Sy) in the case of equal weights, strict internality, and (As)m for all m, the property of substitution. Further {&n(a;w))r =<бп(аг;Ш), г е R. (9) (10) Remark (v) A relation between 0η(α + έ;υ;) and (3n(a;w_) and &n{k;w) is more difficult to obtain and is proved later, III 3.1.3 (10) and is sometimes called Holder's inequality-, see also 1.1 Remark (vi). Remark (vi) A particularly simple case of (9) is : ®n(a 1;w) = (&n(a]wY) Remark (vii) While it is clear that the definition of the geometric mean requires the η-tuple a to be positive, and the definition of the harmonic mean requires that no element of α be zero extensions have been considered; [Asimov]. 1.3 Some Interpretations and Applications 1.3.1 A Geometric Interpretation If 0 < a < b let ABC Ό be a trapezium with AB, DC the parallel sides and AB — b, DC = а, К the intersection of AC and BD, see Figure 1. e/ c, / л уЛг s\ / \ Q If l XX 1 Ij Figure 1 IJ, GH and EKF are parallel to AB; IJ bisects AD and ВС; GH divides ABCD into two similar trapezia. Then IJ = Ща, b), GH = <3(a, b), EF — S)(a, b). The above diagram suggests the following inequality that can be checked in a few numerical cases by the reader; a = min{a, b} < S)(a, b) < 0(a, b) < Ща, b) <b = max{a, b}, (и)

Means and Their Inequalities 67 with equality if and only if a = b. This is an important result, a case of the geometric mean-arithmetic mean inequality; we will return to below in section 2. Remark (i) That EF lies above I J, so that $)(a,b) < 2l(a, b), can be seen as follows: consider what happens to EF as DC increases in length from 0 to a. An extremely interesting series of applications of averages in graph theory and combinatorics is to be found in [Wilf]. 1.3.2 Arithmetic and Harmonic Means in Terms of Errors Given two positive numbers a, b with a < b, and some number χ with a < χ < b, if we guess у to be the value of χ then the error in making this guess is e = \y —x\, while the relative error is ρ = \y — x\/x. How should у be chosen so as to minimize the possible error, respectively relative error? A classical theorem of Cebisev gives a method for such problems: choose у so that the errors, respectively the relative errors, in the two extreme positions are equal. In the first case e = mina<y<i,таха<ж<{, {\y — x\ > occurs for the у such that \у-а\ = \у- b\. That is у = (а + Ь)/2 = Sl(a, Ь). In the second case ρ = тта<у<ь таха<х<ь < \y —x\/x > occurs for the у such that \y - а\/а = \y- b\/b. That is у = 2аЬ/(а + Ъ)= й(а, Ь). In words: the approximation that yields the minimum for the greatest possible value of the error, respectively the relative error, committed in approximating an unknown quantity between two known positive bounds is the arithmetic mean, respectively, harmonic mean, of these bounds. Remark (i) This idea, first found in [Pdlya], has been investigated in detail; see [Aissen 1968; Beckenbach 1950; Metcalf; Mon, Sheu L· Wang С L 1992a). Remark (ii) A similar discussion for a different mean can be found in III 5.1 Remark (i). Given η real numbers а = (αχ,..., αη) we can ask for the value of χ such that the quantity Y^=i(di — x)2 is minimized. Using calculus, or the fact that a quadratic attains its minimum at the mid-point of its zeros, we see that the required value of χ is 2ln(fl)· If the numbers a^, 1 < г < η, are observations that are given weights Щ, 1 < i < ft, then the best possible value for χ is Strifes)· 1.3.3 Averages in Statistics and Probability The use of averages in statistics and probability is too well known to need elaboration; see for instance [Moroney Chap.4), [Norris 1976], and an article by the Moroney in [Newman J. р.Ц57].

68 Chapter II The Italian school of statisticians has made an extensive study of this subject as the many articles in the Bibliography show; see [Barbensi; Bonferroni 1923-4; Gini 1940, 1949, 1952; Gini, Boldrini Galvani, & Venere; Gini Sz Galvani; Roghi], and all the issues of the journal Metron founded by Gini, the most prolific member of this school. If we write α = 21η(α) then the variance of α is 2ln((a — ae)2); the square root of this, the standard deviation more properly fits into the section on power means, III 1 being in the notation defined there Qn(a — ae). Another property of the arithmetic mean is as follows: the normal distribution is the only probability distribution Ρ such that D(x) = ΠΓ=ι P(ai — x) nas its maximum at 2ln(a) ; see [Renyi p.311} and for an extension see [Hosszu L· Vincze}. In this application a need not be positive. Gauss stated that the most probable value of a series of numerical observations was their arithmetic mean and proofs of this from more basic hypotheses about probability have been given by various authors; [Bemporad 1926, 1930; Broggi; Schiaparelli 1868,1875, 1907; Schimmak; Tisserand]. The proof given in [Whittaker & Robinson pp.215-217} was disputed by Zoch although his objections have in turn been disputed, [Wertheimer; Zoch 1935, 1937}. While this statement of Gauss was made as a postulate, the arithmetic mean of any number of observations tends to the true value as the number of observations increases although this is a property of many other means; [Whittaker & Robinson p.215}. Other uses of these ideas are made below, see 2.5.4. 1.3.4 Averages in Statics and Dynamics If the η-tuple a denotes the coordinates on a line of η particles whose masses are given by the η-tuple w, then 2tnfe w) is the coordinate of the centre of mass; %1п(й2]Ш) is the square of the radius of gyration of the system about the origin; the radius of gyration itself is more properly considered as the power mean W[„](a;w) defined in III 1; see for instance [Fowles, Chaps. 7,8}. In this application a need not be positive, in fact if we allow the particles to be situated in space a can be an η-tuple of vectors. 1.3.5 Extracting Square Roots A very ancient use of the arithmetic and harmonic means is Heron's method of extracting square roots; [Heath vol.11 pp.323- 326}, [Chajoth; Pasche 1946, 1948}. Suppose we wish to find the square root of the positive number x; choose any two numbers a, b with 0 < a < b and ab = x. Putting a0 = a, b0 = b define inductively, an = S)(an-i,bn-i), bn = 2t(an_i,6n_i), η e N*. It is easy to check that for all

Means and Their Inequalities 69 η e N,anbn = χ and, using (11), that an < αη+ι < bn+i < bn. η e N. Further , . K-i ~ an-i ^ b-a bn-an < < ■ · ■ < -^-, η Ε N. So linin^oo an = linin-^oo bn = yfx. This result can be expressed as follows: the iteration of the arithmetic and harmonic means of two numbers converges to their compound mean, the geometric mean. The iteration of means and compound means are discussed in more detail later; see VI 3. Remark (i) Heron's method has been extended to roots of higher order; see VI 3.2.2 (e), [Georgakis; Nikolaev; Ory]. Remark (ii) A discussion of this topic and other simple concepts associated with means can be found in [Bullen 1979; Carlson 1971]. Remark (iii) An iterative method for approximating higher order roots by using geometric means, that is, in effect using square roots to obtain higher order roots, has been given; see [Vythoulkas]. 1.3.6 Cesaro Means The famous Cesaro means used to sum divergent series, in particular Fourier series, are examples of arithmetic means; [Hardy 1949, p.96]. Thus, given a sequence a = (α0,αι,...) define Ak,k,j Ε Ν as follows:7 A® = Α,+ι, j e Ν; Α) = Σΐ=ο А*"1» к 6 N*. Then the k-th Cesaro mean of the sequence α is defined by Ak n + k к [Note that if α = ег then Akn = (П ~[ J.] :ulations lead to, ^=7^rE(n"i!i"1)^=a-1^· Simple calculations lead to, 1=0 к a /a \ ~ /n — г -f- /c -f-1\ where A = (Ab .. .),w = ( ],0<г<п. Remember that the sequence starts with ao so, with the notation of Notations 6(xi), An=S^Jl_ αι

70 Chapter II By analogy we can use the geometric or harmonic means in a similar manner; [Pizzetti 1940]. For instance: let G° = ay, j e N; G) = ΓΕ=ι <3*~\ ^N*, and ίθί(α) = (σ*)1/^). 1.3.7 Means in Fair Voting A very interesting application of the three elementary classical means has recently been made to the problem of the fair distribution of seats in the U.S. House of Representatives amongst the fifty states. This problem and this use of these means is discussed in [Sullivan], where other references are given. After this paper the whole area of fair voting has received considerable attention; see for instance [Balinski &; Young; Saari]. 1.3.8 Method of Least Squares Given a set of data r^ = (щ,Уг), 1 < г < η, we can ask for the best line ν = mu 4- с that fits this data in the sense that the sum Y^=1(vi — тщ — с)2 is minimized. That is the squares of the vertical distances of the data points from the line is minimized. This procedure goes under the name of the method of least squares. A simple application of calculus shows that Kn{uv) - Kn{u)Kn{v) %п{и2)%пЫ) ~ 2tnfe)2i m= 2in(M2)-2in(n)2 ' °~ Мм2)-21пЫ2 ' where of course u— (ui,..., un), y_= (vi,..., vn). See [EM5 pp.376-380; Whit- taker & Robinson pp.209-259], [Encke]. 1.3.9 The Zeros of a Complex Polynomial The following are interesting results on the zeros of a complex polynomial. Theorem 7 If f(z) = ao 4- ■ · · 4- anzn is a complex polynomial of degree η, αη φ 0, with zeros z_— (z±,..., zn) and ifV = (z[... . z'n_^) are the zeros of f then 2tnU)=2tn-i(l/)- D Elementary algebra tells us that ζ\Λ l· zn = — (αη_ι/αη). Since f{z) = αϊ 4- · · ■ 4- ηαηζη~1, we have by the same reasoning that z[ 4- l· z!n_1= -((η-1)αη_ι/ηαη). The result is now immediate. D Theorem 8 If f(z) = ao 4- · · · 4- anzn is a complex polynomial of degree η, αη ^ 0, with zeros z_= {z\)..., zn) then for each zero of f, z' say, there are non-negative weights w_ such that z' = 2in (z_; w). D The result is trivial if z' is also a zero of /, and in this case we have all weights except one zero.

Means and Their Inequalities 71 Assume that this is not the case, z' is not a zero of /, f(z') ^ 0. Then: п_.7Ч^У_у7" ι \ у *'-** Solving for У gives the result with weights w^ = — —, 1 < к < η, and the \z' - zk\2 weights are all positive in this case. D This result is known as the Gauss-Lucas theorem and geometrically says: if the zeros of / are the vertices of an n-gon then any zero of f lies inside this n-gon; [Milovanovic, Mitrinovic & Rassias pp.179-183]. Remark (i) For a different result involving the zeros of / and f see 5.6 below. Remark (ii) There are of course many other applications of the arithmetic mean in the literature; [Dome; Jecklin 1971; Martins]. 2 The Geometric Mean-Arithmetic Mean Inequality 2.1 The Statement of the Theorem Although the inequality between the arithmetic and geometric means8 in its simplest form, 1.3.1(11), was probably known in antiquity, the general result for weighted means of η numbers seems to have first appeared in print in the nineteenth century, in the notes of Cauchy's course given at the Ecole Royale, [Cauchy 1821, p.315], although certain results of Newton are related to this inequality, see I 1.1 Remark(ii) and V 2 Remarks (ii), (v). Theorem 1 [The Geometric Mean-Arithmetic Mean Inequality] Given two n-tuples a and w_ then ®n(a]w) <2in(a;^), (GA) with equality if and only if α is constant. This section is devoted to proofs of (GA); proofs giving sharper results that imply (GA) will be given in later sections, or chapters. Corollary 2 Given two η-tuples α and w_ then Йп(а;ш) < ®п(а;ш) < 21п(а;ш), (1) 8 This result is named in various ways; the arithmetic-geometric mean inequality, the arithmetic mean- geometric mean inequality, etc. We will usually refer to it as (GA)

72 Chapter II with equality if and only if α is constant. Π (г) Using the first identity in 1.2(7) and (GA) we have that Йп(а;ш) = (&п(а-1;ш)) < (вп(а_1-,ш)) = ®п(а]Ш)- (ii) Alternatively following Transon, [Transon], we can use the second identity in 1.2(7). This on rewriting and using (GA) gives: l =gn(n?=1|Wfli, 1<^<п;ш) <бп(а;ш) ΠΓ=ια* >^η(ΠΓ=1,^^, 1<з<щш) ^ ι The case of equality is also an easy deduction from that of Theorem 1. D Remark (i) The apparently weaker inequality fin (a; w) < Sln (a; ш), (НА) implies (GA); see 2.4.3 Proof (xix). Remark (ii) If we consider all the η-tuples with given harmonic and arithmetic means we can ask for the range of possible values of their geometric means. In the case of equal weights Post has shown that the extreme values are attained when we choose the η-tuple to have only two values, and one of them is taken (n — 1) times; [Post]. Remark (iii) For another form of (GA) see III 2.1 Remark (vii). 2.2 Some Preliminary Results Before proving 2.1 Theorem 1 we will consider some results that help to simplify later work. 2.2.1 (GA) with η = 2 and Equal Weights We first consider (GA) in its simplest form, 1(11), and give several proofs of this very elementary result; there are no doubt many more. Lemma 3 If α and b are positive real numbers then V^<^, (2) with equality if and only if а = Ъ. Remark (i) First note the interesting fact that when η = 2 the geometric mean is itself the geometric mean of the arithmetic and harmonic means; that is &(а,Ъ) = у/Ща,Ъ)Я(а,Ъ).

Means and Their Inequalities Using this we can write inequality (2) in any of the following forms: 0(a, b) < 2t(a, Ь), £(a, Ь) < 0(a, Ь), й(а, Ь) < 2t(a, Ь). 73 D (г) The result is immediate from the identity (a 4- b)2 = 4ab 4- (a — b)2. This identity is illustrated by Figure 2, in which 0 < a < b. a b b a Figure 2 (гг) The result follows by noting that \Vb — y/a\ > 0 => α 4- b — 2л/аЬ > 0 (Hi) Since (2) is homogeneous9 there is no loss in generality in assuming ab = 1, or equivalently that b = I/a when Lemma 3 is equivalent to: a -\— > 2 with a equality if and only if a = 1, which is just I 2.2 (20). (iv) A variant of the the previous proof is to note that if ab = 1,0 < a < b, then 0 < a < 1 < b and so (b — 1)(1 — a) > 0; expanding this last inequality leads to a + b> l + ab = 2. (v) The first geometric proof is in [Heath vol.11 pp. 363-364; Pappus Book 3, p. 51} and is illustrated by Figure 3. The angles ADC, DEC and DOF are right angles Figure 3 DO=l/2(b-a) ° An inequality P>Q is said to be homogeneous when the function P — Q is homogeneous.

74 Chapter II Take any point D on the diameter AB of a semi-circle of centre 0, let AD = a, DB = b. Construct the right angle ADC, then CD = y/ab, and CO = (a + b)/2. The shortest distance from С to AB is the perpendicular distance so CD < CO with equality if and only if D = О. If DE is perpendicular to CO it is not difficult to check that С Ε = 2ab/(a Η- b) = Д(а, Ь). So using a similar argument to the above some have proved that if a =£ Ь then й(а,Ь) < 0(а,Ь); see [Ercolano 1972, 1973; Gallant; Garfunkel L· Plotkin; Grattan-Guinness; Schild; Sullivan]. (ш) A similar proof can be found in [Ercolano 1972} but using Figure 4. с Figure 4 Angles CON, ONT, OTD are right angles Here AD = b,BD = a; and then OD = (a + b)/2,ND TD = yfob. (vii) Another geometric proof is given in Figure 5. 2ab/(a 4- b), and Figure 5 Let ABCD be a square of side b, and let ABFE be a rectangle of sides α and b. Then area ABFE =area AGE + area ABFG that is <area AGE 4- area ABC;

Means and Their Inequalities 75 which is equivalent to (2). Further equality occurs only when ABC and ABFG have the same area, that is if and only if a = b. Figure 6 (3 21 (viii) The arithmetic mean of a and b is the common value of the co-ordinates of the point where the level curve of f(x, y) = χ + у passing through the point (a, b) meets the line у = χ. This simple observation, together with a similar one for the geometric mean that is obtained when / is replaced by g(x, у) = ху gives a simple proof of Lemma 3 base on the geometry of these curves. Assume that 0 < a < b and let OGA, PGQ, PAQ be the curves у = x, xy = ab, χ 4- у = a-\-b, respectively, see Figure 6. Then G = (Φ(α, b), 0(a, b)) and A= (Ща,Ь),Ща,Ь)). Since the function f(x) = abx~x, χ > 0, is convex, I 4.1 Corollary 7(a), the chord PQ, that is the line χ 4- у = a 4- Ь, lies above the graph of /, I 4.1 Remark (Hi), that is above xy = ab, and so the point G is to the left of the point A; that is 0(a,b) <2t(a,b). Alternatively consider the fact that the two curves PAQ, PGQ only meet at Ρ and Q, and that at Ρ the slope of the first is —1, and that of the second is —a/b. Since — a/b > — 1, the curve PGQ lies to the left of PAQ between Ρ and Q. (ix) The exponential function is strictly convex, I 4.1 Corollary 7(a), so by (J), (χ 4- у ч exp χ 4- expу . Λ Ί. .r ι ι ·г ^ exp ( J < , with equality if and only it ж = у. Putting a = ex,b = ey and noting that the exponential function is strictly increasing completes this proof. (x) A simple proof is given in [Hasto]. If 0 < a < b then putting u2 = b/a then и > 1 and which is not less than 1 by I 2.2 (20). (xi) See 5.5 Remark (ii); [Burk 1985, 1987]. %(q, b) _ Щ1,и) &(а,Ъ) ~ 6(1,и) ~ 2vn D Remark (ii) Proofs (m), (w)and (ix) will be adapted to prove (GA); see 2.4.2

76 Chapter II proof (жг), 2.4.3 proof (xxix), 2.4.2 proof (xvi). Remark (iii) Proof (гг) which is in [Eves 1980 p. 14] can be elaborated to give a further inequality; see VI 2.1.4 Theorem 22. Remark (iv) Proof (x) is an elaboration of proof (iii) and proves a little more; not only is 2l(a, b)/&(a, b) bigger than 1, but if a < b the ratio is increasing as a function of b and decreasing as a function of a. A similar proof can be given by considering the difference 21 (a, b) — Φ (a, b) — —G(u) where G(u) = 1 + 2u(u — 1). 2.2.2 (GA) with η = 2, the General Case We now consider (GA) in its next simplest form, part (a) of the following lemma. Lemma 4 (a) If a, b, a and β are positive real numbers with α -f β = 1 then ааЪР < αα + pb, (3) with equality if and only if а = b. (b) If either а < 0 or а > 1 then (~3) holds, with the same case of equality. D (a) We give eleven proofs of this result. (г) Inequality (3) can be written as (—) < l + a(-— l), which follows from b b (B), I 2.1, on putting a/b = 1 4- x- (ii) It follows from a simple case of 2.2.3 Lemma 5 below that we need only consider a and β rational; the following proof using that assumption is given in [Aiyar]. Let а = p/(p+q), β — q/(p+q),P,q £ N* and, assuming that 0 < α < Ь, subdivide [α, b] into ρ 4- q equal sub-intervals, each of length (b — a)/(p 4- <?), by the points Жг, 0 < 2 < ρ 4- <?, putting xq — a, xp+q = b. Then: Xi -Xi-i = (b-a)/(p + q), 1 < г <p + q, Жг+ι = fe + ^+2)/2, 0 < i < p + q-2, and so by (2) a_ Χχ_ Жр+д-ι Χι Ж2 & Now put с = (α/χι,χι/χ2,... ,xq-i/xq), b= (xq/xq+i,... ,xp+g_i/b). Then from the above and the internality of the geometric mean, 1.2 Theorem 6, 0q(c) < 0p(b); that is φ1/9<φ1/Ρ· Simple calculations show that xa = , which on substituting in the last p + q inequality gives the desired result.

Means and Their Inequalities 77 A similar proof can be given using Xi/xi-ι = (a/b)1^^^^ < г < ρ + q, and y/XiXi+2 — Xi+i to obtain the sub-intervals, (ш) The method of proof (viii) of Lemma 3 can be adapted to this more general situation. The weighted arithmetic mean, respectively geometric mean, of a and b is the common value of the co-ordinates of the point where the level curve of f(x,y) = ax + Py, respectively g{x,y) = xay@, through P(a,b) meets the line у = x. Figure 7 0 21 Let OGA,PGQ,PAQ be у = х,хау(3 = ааЪ?,ах + py = aa + pb respectively, where we assume that 0 < a < b, see Figure 7. Then G = 0(a, b; α, β), A = 2i(a, b;a,/?), and it is not difficult to check that PGQ is convex or that it is to the left of PAQ at A, that is 0(a, b-,α,β) < 2i(a, b;a,β). In this case however we must verify that the two curves meet as shown in a point Q(a',b'). Simple calculations show that the two curves meet in points (x, y) where h{x) = 0, with h{x) = αχχΙβ - {μα + fib)xa^ + paa^b. Clearly h(Q) > 0, h{a) = 0, and for large x, h(x) > 0; further h has a unique negative minimum at χ = 21(α, b). So h has two zeros a, a7, with a < 2l(a, b) < a'. (iv) The strict convexity of the exponential function can be used as in proof (ix) of Lemma 3; see [Mullin]. (v) Consider the function t{x) = αχΡ+βχ~α. Differentiation shows that if χ > 0 then t(x) > t(l) = 1. Substituting χ = a/b, gives (5), and the case of equality.

78 Chapter II (vi) The method used in proof (vii) of Lemma 3 can be generalized. Figure 8 Assume that ΑΒ,ΑΌ are the co-ordinate axes and that AGC is the curve у = xa^, see Figure 8.. Then, as in the proof mentioned, if AB = b, AE = a, area ABFE -area AGE + area ABFG < area AGE 4- area, ABC; (4) or, by simple calculus ab<cm1/Q + /?b1//3; and a simple change of variable completes the proof. Substituting a = l/ρ,β = 1 - 1/p = 1/У, in (4), where p' denotes the conjugate index, see Notations 4, gives: ι. ^ αΡ Ψ ab<— + —, Ρ Ρ' (5) a form of (3) used later; see III 2.1. Inequality (5) is sometimes called Young's inequality, although it is really a very special case of that inequality, [AI pp.48-49; MPF pp.379-389]. (vii) This is another proof of the case of rational weights; [Brown]. Let m, η be two positive integers and 0 < с < d, then cm-l+cm-2d + ■ + dn > > cn-dn сп-цсп-2^+ hd"-1 ndn~x ' ndn On multiplying this gives: ndn(dm — cm) > mcTn(dn — cn). On rewriting this inequality we get mcm+n + ndm+n > (m + n)cmdn. Now in this last inequality put а = cm+n,b = dm+n to get (3) in the case of rational weights. (viii) The restriction to rational weights in the previous proof can be removed, see [Bullen 1997].

Means and Their Inequalities 79 Consider the distinct power functions φι(χ) = xu, 4>2{x) = xv] where и =£ ν, и, ν > 1 and χ > 0 . Applying the mean-value theorem of differentiation, see I 2.1 Footnote 1, to both of these functions on the interval [c, d], с > 0, we get, on cancelling the common factor d — c, du - cu иеил~Х dv - cv ve0 for some ei, ег, with с < ei, e2 < d. Using the last condition and и =£ -у, η, -у > 1, we get that du -cu uc4'1 ucu dv -cv vdv~1 ~vd»' or on multiplying out vdv(du — cu) > ucu{dv — cv). We have assumed that и =£ ν but it is easy to check that the last inequality remains valid when и = v. The proof now proceeds as in proof (vii) with η, ν instead of га, п respectively. If the Cauchy mean-value theorem10 is used we can take e± = e<i and then we need only take u,v > 0, although given the fact that we only really use the ratios u/(u + v), v/(u + v) the initial restriction, u, ν > 1, is unimportant. (ix) The method used in proof (in) of (J), I 4.2 Theorem 12 , can be adapted to give a proof of (3); [Bullen 1979, 1980]. Changing notation by putting χ = α, у = b, t = β, 1 — t = a, (3) becomes G(t) = ж1-'?/ > A(t) = (l-t)x + ty, 0<t< 1. (6) Further since the equality is trivial if χ = у we can, without loss in generality, assume 0 < χ < у. It is easy to see that both of the functions A and G strictly increase, from the value χ when t = 0 to the value у when t = 1. Simple calculations give: A'(t) = y—x, G'(t) = (\ogy — \ogx)G(t)] and A"(t) = 0, G"(t) = (logy- log x)2G(t). Then clearly A' > 0 and G' > 0, which confirms the above remark. Further however G" > 0 so G is strictly convex. Since A is linear and A(0) = (3(0), A(l) = G(l), we see that the graph of A is a chord to the graph of G] and G being strictly convex it has a graph that lies below this chord, which is just (6). An alternative method for this proof is given below in the discussion of (b). The Cauchy, or extended, mean-value theorem states:if the functions f,g are continuous on [a,b] and differentiable on ]a,b[, with g' never zero, then there is a point c, a<c<b, such that ^J_ Г = 4i4; [CE p.598]. g'(c) ' L ^ J

80 Chapter II (χ) The following is a variant of proof (v); [Maligranda 1995] If ρ > 1 and а, Ь > 0 consider the function /(i) = ii-1/p/a+^i1/pb, ί >0, Ρ Ρ' where p' denotes the conjugate index. Simple calculations show that f-(l + l/p') /'(*) = —(Ы~а)> PP so that f(t) > /(α/Ь), with equality if and only if t = α/b, that is αι/ρ&ι/ρ'< Vi/p'a + IiVPb, with equality if and only if ί = α/Ь. Now putting t=\ ,- = a, — = /? in the last inequality gives (3), and the inequality is strict unless a = b. (xi) See also the proof of VI 5 Theorem 2 (a). (b) We divide this part of the lemma into two cases. Case (i) : a < 0 Put a' = αβ, b' = ЪР, a' = -α/β, β' = 1 - a' = Ι/β. Now apply (3) to a7, b', α', β'. Case (ii): a > 1 In this case β < 0 so the previous argument can be easily adapted. Alternatively we could use (B). The proof (ix) of (a) gives both cases immediately. With the notation of that proof write D(t) = A(t) - G(t), t e R. Then we have D(0) = £>(1) = 0, and D"(t) < 0. Hence D(t) > 0, 0 < t < 1, and D(t) < 0, 0 > i,ori > 1. D Remark (i) It follows from proof (i) of (a), and (b) that Lemma 4 and I 2.1 Theorem 1 are equivalent. In particular proofs (ii)—(vi) of (a) can be used to give alternative proofs of (B). 2.2.3 The Equal Weight Case Suffices We now show that 2.1 Theorem 1 can be deduced from its equal weight case. Lemma 5 It is sufficient to prove Theorem 1 for the case of equal weights. D Once Theorem 1 has been proved for constant w, simple arithmetic arguments lead immediately to the case of rational weights; then (GA) for general real w_ follows by a limit argument. To complete the proof it must be shown that if α is not constant then (1) is strict.

Means and Their Inequalities 81 Suppose that not all of the weights are rational and write Wi = щ -f Vi, where щ > 0 and Vi e Q+, 1 < г < η. By (GA), ^nfe^) < 2^nfe^); also since a is not constant we have by Theorem 1 with rational weights, Фп(д;^) < 2ln(&;^)· So ,Un/Wn / NVn/Wn / \Un/Wn / \1 :(Яп(а;м)) («η(α; ν) J < <77r-2tnfe^) 4- 77^-2tnfe^), by (5), =2ΐη(α;«;). D Remark (i) This result seems to appear for the first time in [HLP p. 18]; see also [Herman, Кисета & Simsa рр.ЦЗ, 170-171]. Another proof is given later; see III 3.1.1 Remark (v). The result is generalized in 3.1 Lemma 2. 2.2.4 Cauchy's Backward Induction We now give a famous result of Cauchy, [Cauchy 1821 p.315]. It enables the proof of Theorem 1 to be reduced to the case of η-tuples with η belonging to some strictly increasing sequence, usually η*; = 2fc, к e N*, and is part of Cauchy's proof of (GA), see below 2.4.1 proof (ii). Lemma 6 If Theorem 1 has been proved for a particular positive integer then it is valid for all smaller positive integers. D By Lemma 5 it suffices to consider Theorem 1 in the case of equal weights holds. Assume the result holds when η = m, that k£W,l<k<m and α a positive /c-tuple. Define the m-tuple b by h = ( di, if 1 0i \А=а*(а;щ), if /c < i < k\ к < г < m. By the hypothesis, Фт(Ь; w) < 2lm(b;w/); this inequality being strict unless b is constant. -This inequality is easily seen to be, or ($fc(a) < A = 2Ц(а), and there is equality if and only if α is constant. D Remark (i) It follows from Lemma 6 that to prove (GA) it is sufficient to show that if0<ni<ri2<·--, lim^oo n^ = oo then the assumption of the validity of (GA) for n = Пк implies its validity for η = rifc+i. We can formalize this as the Cauchy principle of mathematical induction; [Dubeau 1991b].

82 Chapter II Theorem 7 IfnoE N* and V{n) is a statement about integers η > no such that: (a) P(no) is valid; (b) ifV(k) is valid for some к > η$ then there is an integer n^ > к such that V(nk) is valid; (c) ifV{k) is valid for any к > п$ then V{k — 1) is valid. Then V(n) is valid for all η >uq. 2.3 Some Geometrical Interpretations Before turning to the proofs of (GA) we give some particularly simple forms of that inequality that have interesting geometrical applications. Lemma 8 Theorem 1 is equivalent to either of the following. (a) If α is an η-tuple such that ΠΓ=ι α* — 1 then Σ™=1 оц > η, with equality if and only if α is constant; (b) If α is an η-tuple such that Σ7=ι аг = ^ then ΠΓ=ι α* — (Vn) > with equality if and only if α is constant. D In both cases one implication is trivial. Assume that (a) holds ; let α be any positive η-tuple; define b= (■^")···)"ί" · η η Clearly Y\bi = 1, and so by (а) /__]Ьг > п. This by the definition of b is just i = i i=i % > (δ. Assume that (b) holds ; let α be any positive n-tuple; define с = ( ——,..., —^-). Vn2i nvx) η η Clearly /_^Q = 1, and so by (b) TTq < (l/n) . This by the definition of с is just (3 < SI. Thus either (a) or (b) is sufficient to imply (GA) in the case of equal weights, which by Lemma 5 is sufficient to prove this lemma. D Remark (i) The case η = 2 of Lemma 8(a) is just I 2.2 (20); for (b) see [Goursat]. Both results are classical and proofs can be found in many places; see for instance [Chrystal pp.52-56], [Darboux 1887]. Both parts of this lemma have simple geometric interpretations. Corollary 9 (a) Of all η-parallelepipeds of given volume the one with the least perimeter is the n-cube. (b) Of all η-parallelepipeds of given perimeter the one with the greatest volume is the n-cube. Corollary 10 Of all the partitions of the interval [0,1] into η sub-intervals, the

Means and Their Inequalities 83 partition into equal sub-intervals is the one for which the product of the intervals is the greatest. See [Polyа 1954 P-129]. Lemma 11 (a) In the case of η = 3 (GA) is equivalent to the statement: of all the triangles of given perimeter the equilateral triangle has the greatest area. (b) In the case of η = 4 (GA) for Α-tuples of numbers the sum of any three of which is greater than the fourth is equivalent to the statement: of all the concyclic quadrilaterals of given perimeter the square has the greatest area. D (a) Let а, Ь, с be the lengths of the sides of a triangle, s = (a 4- b 4- c)/2 its semi-perimeter; then by a formula of Heron, [Heath vol.II pp. 321-323; Melzak 1983b pp. 1-3], its area is A = y/s(s — a)(s — b)(s — c). When the triangle is equilateral this area is AQ = s2/Зл/З. By (GA) in the case η = 3 and equal weights, Л = ^(а-а)(а-Ь)(а-с))3/а<^(а-а) + (азЬ) + (а-с))3/2 with equality if and only if s — a = s — b = s — c, or a = b = с Conversely if 01,02,03 are any three positive numbers define a, 6, с by a\ — s — a,a,2 = s—b, as = s — c, where s, as above, is (a4-b4-c)/2 . Then simple calculations show that s = a\ 4- 02 4- 03 so a = s — a\ = 0,2 4- 03, b = 03 4- a±, с = a\ 4- 02 are positive and further a + b — c, b + c — a,c-fa- b are also positive, being 2аз, 2αι2α2 respectively, so that а, Ь, с are the sides of a triangle. Now using the inequalities above /Л\2/3 /Л0ч2/3 03(αι,α2,α3) = ^—/=J < ( "7=) = S/3 = ^з(<Н, «2, аз) with equality if and only if a = b = c, or a\ = а2 = аз. (b)Let а, Ь, с, d be the sides of a concyclic quadrilateral, s = (а4-Ь4-с4-а1)/2 its semi- perimeter; then, by a formula of Brahmagupta, [Melzak 1983b pp.4-6], its area is A = y/(s — a)(s — b)(s — c)(s — d). When the quadrilateral is a square this area is Aq = s2/4. Noting that the 4-tuple a\ = s — a, a2 = s — Ь, аз = s — c, a4 = s — d has the property required in the hypotheses— the sum of any three is greater than the fourth— apply the equal weight (GA) in this special case to get 2 ((s - a) 4- (s - b) 4- (s - c) 4- (s - d\ ^(s-a)(s-b)(s-c)(s-d))2<(

84 Chapter II with equality if and only if s — a = s — b = s — c= s — d, or a = b = с = d. Conversely if a\, a2, a^, (14 are any four positive numbers satisfying the hypothesis that the sum of any three is greater than the fourth define, as in (а), а, Ь, c, d by a\ = s — a,a2 = s — b,a,3 = s — c,a4 = s — d, where s, as above, is (a + Ь-f c + d)/2. Then simple calculations show that 2s = a\ -f a2 4- as 4- &4 so 2a = 2s — 2ai = a2 4- 03-4-04 — αϊ > 0 and similarly b, с and d are positive. Further a+b+c—d = 2а^ > 0, b + c-\- d— a = 2αι >0,c-fii-fa-6= 2a2, d -f a -f- Ь — с = 2аз so that а, Ь, с, d are the sides of a coneyclic quadrilateral, see [Melzak 1983b pp.8-9). Now using the inequalities above 04(αι,α2,α3,α4) = A1/2 < Ay2 = s/2 = 2s/4 = 214(>ι, α2,α3,α4), with equality if and only if a = b = с = d, or a\ — a2 = аз = a±. Noting that Lemma 5 shows that (GA) for equal weights and a given n-tuple implies the general case for the same η-tuple completes the proof. D Remark (ii) Extending this to a general concyclic n-gon seems difficult as there seems to be no simple formula for the area of a concyclic n-gon, η > 5; see [Robbins]. The need to phrase part (b) of this lemma in a manner different from part (a) was communicated to me by Mowaffaq Hajja. Remark (iii) For further discussions of the results in this section the reader should consult the following: [Kazarinoff 1961a pp. 18-58; Kline p. 126), [Bioche; Garver; Shisha; Usai 1940b]. 2.4 Proofs of the Geometric Mean-Arithmetic Mean Inequality The proofs will, as far as is known, be in the order of their appearance. Seventy four proofs are given and there are undoubtedly as many more in the literature. A survey of proofs given up to 1904 can be found in [Muirhead 1901/04) and a survey of several proofs also occurs in [Colwell L· Gillett]. It is sufficient by Lemma 5 to give a proof for the equal weight case and many proofs do this. Also, given 2.2.1 Lemma 3, or 2.2.2 Lemma 4, it is sufficient to give the inductive step in any proof by induction; see also 2.2.4 Remark (i). Further for a complete proof of the theorem it is clearly sufficient to prove that the inequality is strict for non-constant n-tuples. In addition in an inductive proof we can also assume that no two elements of the the η-tuple α are equal, for if two are equal the inequality follows by the induction hypothesis; in fact we may assume that the η-tuple is strictly increasing. The proofs are divided into sections determined by the publication dates of the four main references [HLP; BB; AI; MI), 1934, 1965, 1970 and 1988, respectively,

Means and Their Inequalities 85 preceded by the prehistoric proofs. A baker's dozen of proofs that are in journals not seen by the author are listed for completeness in section 2.4.7. 2.4.1 Proofs Published Prior to 1901. Proofs (i)-(vii) (i) Maclaurin circa 1729 This is by far the earliest proof and it appears to be due to Maclaurin who states the result in the form of 2.3 Corollary 10. D Suppose that 0 < a\ < a2 < · ■ · < an, a\ ^ an. If ai and an are replaced by (αϊ 4- αη)/2 then 21 is unchanged, but using (2) it is easily see that 0 is increased. If then a is varied so as to keep 21 fixed, and of such η-tuples gf is the one at which 0 assumes its maximum value, the above argument shows that g/ must be constant. Hence the maximum of 0 is attained when all the terms of α are equal, and this maximum value is equal to 21. D [Chrystal p.47}, [Grebe; Maclaurin]. Given the date of the proof it is not surprising that the existence of an gf at which (3 attains its maximum was taken for granted. This missing step can be supplied in either of the following ways, [HLP p. 19, footnote(a)]. (a) Use the fact that a continuous function, φ, defined on a compact set, K, attains its maximum on that set. In this situation if χ = (ж1}... , жп), Φ(x) — ( Τ7 χι) ; Κ = Ι χ; Χ{ > 0, 1 < г < η, and V_] x% = η% f· i=i i=i (b) After к steps in Maclaurin's proof let us denote the resulting n-tuple by gSk\ and assume it is increasing. Then clearly a(*) <a(*+D <...<a(W)<aW Hence Нт^оо af, and lim^^oo a^ both exist, say with values a, A respectively. It is not difficult to see that after η steps that maximum difference in the sequence has been reduced by at least one half; that is a(n) _ a(n) < a"~ai and so lim^oo {an — (A ) = 0, or A — a. This argument is due to Hardy. (й) Cauchy 1821 This elementary proof depends on a sophisticated induction argument, see 2.2.4 Lemma 6, and consists of proving (GA) for all integers η of the form 2fc, к £ Ν*.

86 Chapter II D Assume the result is known for к = m and let a = (αϊ,..., α2™), b = (b\,..., b2m) and с = (ci,..., c2m+i) = (a, b) = (αϊ,..., a2™, bi,..., b2m). Now 62Tn+l(c) =7<52т(а)02т(Ь), <y/2i2m(a)2i2m(b), by the induction hypothesis (7) <2t(2t2m(a),2t2^(S)), by (GA) in case η = 2, Lemma 3, (8) =2i2m + l(c). Equality occurs at (7), by the induction hypothesis, if and only if a, and b are constant, α and b respectively say; and then at (8) there is equality if and only if a = b\ that is we get equality if and only if с is constant. D [Cauchy 1821 p.315; Polya & Szego p.64], [Chajoth; Forder). Remark (i) A variant of this proof can be found in [Boutroux]; Bellman has given a particularly lucid version, [Bellman 1954]. (in) Liouville 1839 D Assume (GA) for all integers less than η and put α = (αϊ,..., αη_ι, ж), and 1еЬ/(х)=5^(о)-в^(о)· Then . /77 — 1 γ·\Π —1 f'{x) = %nn-\a) - <δηηΖΐ(α) = (-—^(a) + -) - й5^(а). This shows that f is increasing; further f vanishes when χ = χ', where χ' = η6η_ι(α) — (η — 1)21η(α). Thus / has a unique minimum at x'', and the value of this minimum is f(x') = (η—1)0^ΐ];(α)(21η_ι(α) — (δη_ι(α)), and by the induction hypothesis f(x') > 0, . Hence / > 0, which completes the proof except for the case of equality. In order that f(x) =0 we need both that χ = x' and f(x') =0. By the induction hypothesis f(x') = 0 if and only if αϊ = · ■ ■ = αη_ι = α, say; but then χ' = α and the proof is complete. D [Liouville]. Remark (ii) This early proof does not seem to have been noticed until it appeared in the book by Mitrinovic & Vasic, [Mitrinovic & Vasic pp.28-29]. As a result many other authors discovered it independently; see, for instance, [Buch; Lawrence; Yosida]. Like proof (г) this is not an elementary proof. The method can be used to obtain more precise results; see below 3.1 Theorem 1, and [Ruthing 1982].

Means and Their Inequalities 87 (iv) Thacker 1851. This inductive proof is based on (~B). K{a)=anAl + a2 + '"an + {l~n)ai)\ V na1 J - αι (α2+η-ΪαηΤ~^ ЬУ l 2·2(11)' ΟΓ ЬУ (~B), I 2.1 Theorem 1, with α = -^—, 71—1 >ai^n-ife) = ^nfe)5usmg the induction hypothesis. Further, from I 2.1 Theorem 1 and the induction hypothesis this inequality is strict unless a is constant. D [Thacker]. (v) Hurwitz 1891 D If / is any function of η variables put Ρ(/(αι,..., an)) = ^/(a^, · ·· iain)· So for instance: if /(al5... , an) = axa2 · · · an then P(a1a2 · · -an) = nl^a^ · · · an) = n\<3n(an); if /(ttl,..., an) = < then P«) = (n - 1)!« + < · · · + О = n!2in(an). Define 0fc, 1 < /c < n — 1, by 0fc = P((a7~fc — a^~k)(a1 — α2)α3α± ■ · -α^+ι). Then, as is easily seen, 0^ = P((a™_fc_1 4- · ■ ■ 4- a1^~k~1)(ai — α2)2α$α± · · -α^+ι), and so if α is positive and not constant, фк > 0,1 < к < η — 1 while if α is constant then 0fc = 0,1 < к < η - 1. Also 0, = 2 ^Р(оГН1аа · · ■ ak) - P(an1~ka2 · · · ak+1)\ and so, summing over к we get Υ^ζΐ\ фк = 2(Ρ(α™) — Ρ(αι · · · αη)). Hence, from the initial comments, 1 η—ι 2l„(fin) - 0n(a") = 5πγΣ^>0, 2(n!) £i with equality if and only if a is constant. D [Hurwitz 1891). Remark (iii) This proof is interesting in that it is the first to give an exact value for the difference %n(an) - ®n(an). Remark (iv) In [BB p.8} it is pointed out that this proof contains the germ of a technique that Hurwitz was to use later in his famous paper, [Hurwitz 1897],

88 Chapter II on the generation of invariants by integration over groups. For further discussion along these lines the reader is referred to the papers [Motzkin 1965a,b]. (vi) Crawford 1900 This is a variant of proof (г), more sophisticated but elementary. D As in proof (г) let the η-tuple α be such that 0 < a\ < · · · < an, a\ =fi an and define a/ by changing a\ to 5tn(a) = 21, and an to a\ 4- an — 21, the rest of the a'i = ai. Then 2ln(a') = 21 and η—ι 0£(α') =β^(ο) - Π *(* - «Ж*» " a) >^n(«), the second term being positive by internality. After at most (n — 1) repetitions of this process we arrive at a constant n-tuple a", and this gives a proof of (GA). D [Bnggs & Bryan p. 185; Hardy 1948 p.32; Polya 1954 V-Wl Sturm p.3], [Crawford; Fletcher; Muirhead 1900/01, 1901/1904, 1906]. Remark (v) A similar proof has been constructed by defining a[ = 0n(o) = 0, and an = a1a2/^6> The idea in this proof has been as a basis of a general method of forming inequalities, [Keckic]. (vii) Chrystal 1900 D Again let the n-tuple a be increasing and non-constant, and assume (GA) for all integers less than n. αί / \ n_1oi / \ , an or / \ , «η-2ίη_ι(α) 2ίη(α) = 21η-ι(α) Η = Sin-1 (α) Η —. (9) η η η By internality an — 21η_ι(α) > 0 and so from the right-hand side of (9) К(&) >К-Лй) + (an ~ Sln-i(a))SGli(a). by (~B), =an%nn-_\{a) >an6^lj;(a), by the induction hypothesis, [Chrystal vol.II], [Muirhead 1901/04; Oberschelp; Popovic; Wigert]. Remark (vi) If the inductive hypothesis is used on the middle expression in (9) then we get oi*v W Λ* '~\ ι Дn-^n-l(д)^7г

Means and Their Inequalities 89 and the above argument again gives (GA); see [Weber pp.689-690), [Tweedie]. 2.4.2 Proofs Published Between 1901 and 1934. Proofs (viii)-(xvi) (viii) Muirhead 1900/1901 See V 6 Remark (iii). (ix) Dougall 1905 Following ideas of Muirhead, Dougall made a study of various identities from which (GA) becomes apparent. In particular he gives the following proof of (GA). D With the notation of I 1.1(1) and Example (i) , if α is a non-constant n-tuple the polynomial has distinct roots so inequality I 1.1(4) is strict, that is Л/г ^ jl/H-1 -, ^ ^ <C_r >dn/_r_11 1 <r<n. In particular dn-1 > dQ, which is (GA). D [Dougall]. Remark (i) In fact Dougall gives an exact formula for d™ — dn\ such a formula also occurs in [Jolliffe; Muirhead, 1900/01]. A similar proof can be found in [Green]. (x) Polya 1910 D If α is a non-constant η-tuple define b by a^ = (1 4- M2ln(a; ш)> 1 < i < п. Then b is not the zero η-tuple and Σ™=1 Wibi =0, and ®n (а; Ш) =%n (a; w) ®n (1 + b; w) η <2ίη(a; w) exp ( /^ги^ЬЛ, by inequality I 2.2(8), =Stn(a;ti;). D [HLP p.103], [Alexanderson], [Lidstone; Wetzel]. Remark (ii) This proof is in [HLP] but is assigned this date by Alexanderson. The equal weight case was rediscovered by Lidstone. Remark (iii) An extension of the idea in this proof is given below in 2.4.5 proof (xlii).

90 Chapter II (xi) Dorrie 1921 D If α is a non-constant η-tuple, η > 3, and ΠΓ=ι ai = 1> tnen at least one element is bigger than 1 and at least one is less than 1; assume аг < 1 < αι. Then η η 22 °"i =α1 + α2 + ζ2 ai i=l i=3 η >1 4- αια2 4- Ύ2αΐ> see ^-1 Lemma 3 proof (iv), i=3 >1 4- (n — 1) = η by the induction hypothesis, and by 2.3 Lemma 8(a), this is equivalent to (GA). D [Dome pp.37-39; Korovkin p.7], [Ehlers; Heymann; Kreis 1946]. (xii) Carr 1926 D The following is a simple inductive proof. β»(α) = (вУ-^аГ2/"-1)17""^^:^'1-^) <^;β^η-1(οχ-2/η-1 + ^|βη_ι(α), by (5), the case η = 2, 1 η — 2 η — 2 <7 7То0п(а) + τ ^2αη 4- -(5η_ι(α), by (5) again, (η — Ι)2 [η — 2γ η — 1 1 η — 2 η — 2 <7 ττ?^η(α) 4- τ 7^αη Η 7&η-ι(α), by the induction hypothesis, (η — iy (п — 2)г η — Ι ώ^+Ι1-^)^· (η-1)2 ην^ ' V (η-Ι)2 Simple calculations leads to 0η(α) < 51η(α). □ [Carr]. (жги) Steffensen 1930 We first have the following simple lemma. Lemma 12 If a, b are increasing η-tuples and α < b, then (Y^=1a>i)(J2™=1bi) is not decreased by interchanging dk and b^; further it is increased unless either o>k =bk, or for all i ^ k,di = bi. D This is immediate from the identity η η (Σαί + bk) (Σbi + ak) ъфк г^к ТЪ ТЪ ТЪ ТЪ =(Σα0(Σ^) + (^-α,)(Σ^-Σ^)· г^к г^к

Means and Their Inequalities 91 D Using Lemma 12 we have the following proof of (GA). D Suppose that a is non-constant increasing η-tuple, then nn^{a) =(ai 4- · · · 4- αι)(α2 Η h α2) ... (αη Η l· αη) <(αι 4- α2 Η h αη)(αι 4- α2 Η h α2) ... (αϊ 4- αη Η h αη). by Lemma 12. The result follows by a repetition of this argument. D [Steffensen 1930, 1931]. (xiv) Nagell 1932 D Assuming that α is a decreasing η-tuple define Δ = η(21η(α) — Φη(α)) and A{ = a\'n - аУп, 1 < г < n. Then -£*-^(j)((Gn:i)-Gn:i)£^) + Σ)(Σ<-*Π^)) j fc=l k=l / У^ a^1-·7^71 (-) ( . 9 J Σ ^i' using induction on the terms of У) J=2 J=2 >0 D [iVagell; Solberg]. (ату) Hardy, Littlewood L· Polya 1934 D Suppose that (GA) has been proved for integers less than n, then < Рп-1(а;ш) + цГап, by (5) <—γ—21п-1(й;ш) 4- ^Γ^-αη, by the induction hypothesis, =51п(а;ш), The case of equality follow from 2.2.2 Lemma 4 and the induction hypothesis. D [HLPp.38].

92 Chapter II Remark (iv) A similar proof can be given starting with 5ln(a;u;)· The proof is an adaption of a proof of (J); see I 4.2 Theorem 12 proof (г). Remark (v) Other proofs of (J) and of the Jensen-Steffensen inequality can be adapted to give a variant of this proof; see proof 2.4.5 (Hi) and [Magnus]. Remark (vi) Even in the case of equal weights appeal must be made to (5) rather than (2). However in the equal weight case the appeal to (5) can be avoided by using the the inequality 1.2 (8); see [Georgakis]. (xvi) Hardy, Littlewood L· Polya 1934 D Using the strict convexity of the exponential function we have: ®п(а]Ш) =exp (On(logа;ш)), ЬУ !·2(8) <2ln(exp ο log a; w), by (J), see 1.1(4), =2ln(a;u>). D [HLP p.78], [Barton; Flanders]. Remark (vii) A similar proof can be given using the strict concavity of the logarithmic function, [Gentle 1977; Steffensen 1919]. 2.4.3 Proofs Published Between 1935 and 1965. Proofs (xvii)-(xxxi) (xvii) Bohr 1935 xv D Consider the expansion of exy and in powers of y. Quite trivially no k\ coefficient of the first expansion is less than the corresponding coefficient of the second. It is then quite easy to see that the same is true of the expansions of n (ЕР- Xi)kynk the two functions exp (y V^ xA and -—г~\ ,N , both obtained as products of ν /_^4 / (k\)n functions of the above type. Comparing the coefficients of ynk in the two equations we get that Яп(зО > lf(nk)\\1/kn 0п(ж) - n\(k\)n Now, using Stirling's formula in the form ml ~ т7Пе~тл/2пт, the right-hand side tends to 1 as к —>> oo. D [Bohr]. Remark (i) Bohr's proof does not give the case of equality, but the following simple argument by Dinghas completes the proof; [Dinghas 1962/3]. Consider

Means and Their Inequalities 93 the non-negative function d(x) = 51п(ж) — ®n(x) then dd/dxi = (1 — <3n/xi)/n, 1 < г < п. Now assume that for a non-constant χ we have d(x) = 0. Choose г so that Xi = minx, then dd/dxi is negative and so if X{ is increased slightly d becomes negative, which is a contradiction. (xviii) Dehn 1941 A particularly simple geometrical proof has been given by Dehn based on some observations about angles and chords in a circle. In particular if the angles c^, 1 < г < η, are not all equal and from the centre of a circle cut off chords of lengths Q>i, 1 < i < η respectively then %ln(a) is less than the length of the chord cut off by the angle 2ln(<*); for full details the reader is referred to the reference. [Dehn]. (xix) Walsh 1943 We first prove (HA). Lemma 13 If a and w are η-tuples then fin (а; Ш) < &n (a; ш), (НА) or equivalently i=i ' ^ i=i г ' (10) with equality in either inequality if and only if α is constant. D The proof is by induction, the case η = 1 being trivial. The left-hand side of (10) is equal to П—1 \ •Π—1 \ Σ^+ω«α«)(Σ^ + ^) n_1 / \ >W%_1 4- wn ^2 wi ( — Η ~ J + wni ^ tne induction hypothesis, \ CLi anJ /2 ι On., ТЛ7 ι Λ.,2 >^_, + гшпИ^-х + w2n, by I 2.2 (20), =W2 The case of equality is easily obtained. D For another proof of (10) see III 2.2 Example(ii). Now we proceed to Walsh's proof of the equal weight case of (GA) using (10). D Assume that a is not constant.

94 Chapter II It is easily seen that η / n \ / n \ 4ΣΣ(αΓ1-α?'1)(^-^)>ο. 2 j = i г=1 Hence and by the induction hypothesis η η Σ aTx > (η -!) Πα*'* - fc -η' ъ^к i^k with at most one of these inequalities not being strict. Adding these η inequalities and dividing by η gives |>Γ'>(ίΗ(έέ} Multiplying (11) and (12) gives έ<>Κπ-)(Σ>)(έέ) η >n Y\ai, by the equal weight case of (10). This completes the proof of the equal weight case of (GA), and the case of equality is easily obtained from that in (10). [Walsh]. (xx) Nanjundiah 1952 D For given η-tuples а, w_ define the η-tuples Ь, с as follows: ТТЛ ТТЛ Wi/Wi °% = —o>% ai-i, (H = , , 1 < г < n, Щ Wi а™_г{1/Шг where wq = 0 and ao = I. Then by the symmetric form of (~B), I 2.1(3), с > Ь, with equality only if a is constant. Easy calculations show that 21^(6; w) = a^ = (S>i(c; г/;), 1 < г < п. Hence, using the monotonicity of the arithmetic mean, 21п(с;ш) - ®n(c]w) > 2ln(b;w) - 0n(c; w) = 0.

Means and Their Inequalities 95 This completes the proof since clearly we can define a from c, so с can be an arbitrary η-tuple; the case of equality is immediate. D [Nanjundiah 1952). Remark (ii) This method is the basis of other important inequalities; see 3.1 Theorem proof (v) and 3.4. (xxi) Jacobsthal 1952 This is a simple inductive proof that uses the polynomial in I 1.2(a); the same method is used later for a stronger result; see 3.1 Theorem 1, proof (г). D > n~lV~ ( ( y n ~ . ) -f η — 1 ), by the induction hypothesis, η V 0η-ι(α)/ Π [Climescu; Jacobsthal]. (xxii) Devide 1956 D If α is a non-constant η-tuple define the η-tuple b by a^ 4- fy = 2ln(a), when Вn = 0. The following identities are easily deduced. Br{br+1 4- · · · 4- bn) = -Bl 1 < r < η - 1, (13) and order α to obtain the following properties: Bn=0; M2<0; Wr+i<0,2<r<n-l. (14) This is possible since from (13) not all Brbj, r + 1 < j < n, can be positive. The following identities can also be checked: Brbr+1 = Ып(а) - Br\ar+1 - 2tn(a)i2tn(a) - Br+\ 1 < r < η - 1, and so n+i 5rbr+12i;_1(a) Д <n = Cr - Cr+1 1 < r < η - 1, i=r+2

96 Chapter II where n+i an+1 = 1, and Cr = 2ΐ;_1(α)(21η(α) - £r) Д a», 1 < r < η - 1. i=r+i Adding over these identities gives n—1 n+i r=i г=г+2 and the right-hand side is negative by (14). D [Devide]. Remark (iii) This last identity should be compared with that in 2.4.1 proof (v). Remark (iv) A η-tuple b is the same as χ used in 2.4.5 proof (xxxviii). (xxiii) Blanusa 1956 If a is an increasing η-tuple then (GA) is equivalent to η—ι г (^(n-i)Aai)n>n"n(EAa«)' (15) г=о i=o j—o where Δαο = a\. The case η = 1 of this inequality is trivial. So assume (15) holds as stated, with η > 1. Then, since η—ι η η г (^(η+1)(η-ί)Δαί)"^(η + 1)ηΔαί>ήη+1(η+1)η+1Π(^Δαί), г=о г=о г=о .7=0 it suffices, for the induction, that is to show (l9) holds with η replaced by (n 4-1), to prove that η—ι η η , Γ^(η+1)(η-ζ)ΔαΛη^(η + 1)ηΔα^ < (^η^ + Ι-ήΑαΧ . (16) г=о г=о г=о Put β = ^^(п + 1)(п - ί)Δα,ί, η = Σ"=0 гЛа* and n a = a(j) = 2_](n ~^~ 1)п^аг — J7> 0 < J < n·

Means and Their Inequalities 97 Then noting that (a — j)(P 4- 7) > a/?, we get a(j + 1) Multiplying these inequalities gives α(η)(/? + 7)η >α(0)/Γ, which is (16). D [Blanusa]. (xxiv) Bellman 1957 This proof uses the methods of dynamic programming. D Consider the problem of maximizing ΠΓ=ι α* subject to the condition that Y^i=1 аг = a; see above 2.3 Lemma 8(b). Let this maximum be g(n; a). If we pick an > 0, it remains to obtain the a^ > 0,1 < г < η — 1, that maximize ΠΓ^ι1 ai subject to the condition Y^I^ ai = а — an. Then g{l\ a) = a and if η > 2 д{ща) = maxo<an<a {&п9{п — l;a — an)}. Putting a^ = ab^, 1 < г < η, leads to g(n; a) = ang{n; 1); and (n _ 1 \n-i g(n; 1) = 5(n - 1; 1) m^ {j/(l - у)""1} = ^—J 5(n - 1; 1). Hence, using p(l;l) = 1 we get that д{щ a) = (α/η) , which completes the proof. D [MPF pp.697-708; BB p.6], [Bellman 1957; Mitmnovic & Vasic p.25], [Bellman 1957a,b; Dubeau 1990b,c; Iwamoto; Wang С L 1979a-d, 1980a, 1981a,b, 1982, 1984a). (xxv) Mitrinovic 1958 D Let α be an (n+ l)-tuple and assume (GA) for integers less than n-f-1. Put αη+ι + (η-1)%η+ι(α) / ™η-ι(Λ1,η then by the induction hypothesis, A > G. Further 2ln+i(a) = ~ > \A42tn(tt), by the induction hypothesis, >γ(76η(α), by the above observation, and the induction hypothesis, l/2n {КХ1(а)К+1^ш))

98 Chapter II On rearranging this completes the proof. The case of equality is immediate. D [Mitnnovic 1964 PP-S32 233]. (xxvi) Climescu 1958 D Put χ = a/b in I 1.2(7) to get: nbn+1 + an+1 >{n + l)bna, (17) with equality if and only if a = b. Now let α be an (n + l)-tuple, then <^ГЧй) + ^«Ж,Ъу(17), η Ι < 21η(αη+1) Η τα^ίι"5 by the induction hypothesis, ~n-f-l w ; n+ 1 n+1' J JF =2in+i(an+1). The case of equality follows easily. D [Climescu]. (xxvii) Newman 1960 This is a proof of 2.3 Lemma 8(a). D Let α be a non-constant (n + l)-tuple with ΠΓ=ίι аг = 1· Since not all the elements of α are equal, we may suppose without loss in generality that αη+ι φ 1. Then , n+l η .j/ V^ di >n \\\o'i) + &n+i 5 by the induction hypothesis, i=l i=l Π 4- αη+ι l/n αη+1 >n +1, by (~B) with α = —l/n and 1 -\- χ = αη+ι [Newman D J]. (xXViU) DlANANDA 1960 This is a modification of Cauchy's proof, 2.4.1 proof (гг). D

Means and Their Inequalities 99 D 0„(a) = V^-i(«)«n/n_1®r2/n_1(«) <|(e„_i(fi) + аУ"-1©^-27""1!»)), by 2.2.1 (2), (n = 2 equal weight (GA)), <J(st„_i(a) + -^T + ^en(o)), by 2.2.2 (5), (n = 2 (GA)), and the induction hypothesis, which on simplification gives the result. D [Diananda I960]. (xxix) Korovkin 1961 In Korovkin's book I 2.2 Lemma 6 is proved using (GA). We use the lemma to give a proof of (GA) in the form of 2.3 Lemma 8(a). D Let α be an η-tuple with ΠΓ=ι αΐ = 1 an<^ se^ xk = ΠΓ=&α*' 1 — ^ — n· Then the left-hand side of I 2.2 (24) is just Σ™=1 cli and so by that inequality ΣΓ=ι α* — n as na<^ ^° ^e Pr°ved. D [Korovkin p.8]. (xxx) Mohr 1964 D Let α be a non-constant increasing η-tuple and first we introduce some notation: a<°> = a; a^ = (a^)(1) = (a^\ .. ., al1}), where af} = 2in-i(a0, where a^ is defined in Notations 6(v); and having defined gS°\ ... g^k~^ then a(*) = f a(*-D In other words the terms of a^ are the arithmetic means of the terms of a^-1^ taken (n — 1) at a time. The following identities can easily be established: 2tn(a(/e)) = 2in(a^), к = 0, 1,...; < 2tn(a(0)) + f""^, A; = 1,2,..., by internality. (n — 1)K Again by internality, βη(α(*>) < 2ln(a(0)) + ^3^, к = 1,2,.... (18)

100 Chapter II Now assume (GA) for all integers less than n, and we have immediately that af^en-iife"-1»);) = (вп(Д(*-Ц))П/П-1(в<*-1»)-1/("-1),* = 1,2>.... On multiplying these inequalities we obtain <Sn(a{k)) > «„(a^-1»), fc=l,2,.... (19) In addition since a\ =fi an, ®n(a{1)) > ©n(a(0)). From (18) and (19), Ma(m)) < Sl„(a(0)) + £luk> k=m,m + l,.... (20) Using (18), choose a /c so that β-(α(1)) - ^E^ > *»(Л This last inequality and (20) with m = 1 completes the proof. D [MoJhr 1964]. Remark (v) In fact this proves a little more: 0n(a(m))<2ln(a), m = 0,l,... (xxxi) Beckenbach & Bellman 1965 This is another calculus proof of 2.3 Lemma 10(b). D The object is to find the minimum of the function Σ™=1 αι on the compact set {α;α>0, Π?=ια< = 1}· Using the Lagrange multiplier11 approach, consider f(a) = ΠΓ=ι α* ~ ^ΣΓ=ια*> when —— = TT en — λ, 1 < 7 < η. bo it -— = · · · = —— we must have di = · · · = an. oai oan If / and g, are functions of η variables, and we wish to find the extrema of / subject to condition <7=0 the auxiliary function f (x_,\) = f (x_) + Xg(x) of (n+1) variables is introduced and the problem reduces to finding the turning points of /. The extra variable λ is called a Lagrange multiplier and the procedure is called the method of Lagrange multipliers; [CE p. 1015; EM5 p. 336]

Means and Their Inequalities 101 This gives η as the unique minimum of Σ™=1 &% and proves 2.3 Lemma 10(b). D [BB p.5], [Amir-Moez; Rodenberg]. 2.4.4 Proofs Published Between 1966 and 1970. Proofs (xxxii)-(xxxvii) (xxxii) Dzyadyk 1966 D The following two identities are given by Dzyadyk: η6η(α)+αη+ι gn(q) /6n+i(a)Y ^®n+i(a)H Π~Μ ^ / ч 1; (21) η 4-1 ww n + l^V ^n(a) ^(^-eniaj^^e^xiajpfc-if-^^), (22) where pn is the polynomial of I 1.2(a) . By I 1.2(6) the identity (21) implies that η6^^) + α^+1 > £>η+1(α) and, by the induction hypothesis, the left-hand side of this expression is not greater than —^-=——^— = 2ln+i(a). This proves (GA), and for equality we must have ax = · · · = an, by the induction hypothesis, and 0n(a) = <Sn+i(a), by I 1.2(7). This implies that a\ — · · · = αη+ι. Identity (22), again using I 1.2(7), immediately implies (GA) and the case of equality. D [Dzyadyk]. (xxxiii) Guha 1967 Guha uses the following lemma to give a simple proof of (GA). Lemma 14 If ρ > <? > 0, χ > у > 0 then (px + y + a)(x + qy + a) > ((ρ + l)x + a) ((q + l)y + a), with equality if and only if χ = у. D This is an immediate consequence of the identity (px + У 4- α)(χ + qy + a) - ((ρ + 1)ж + α) ((<? 4- 1)2/ 4- α) = (рж - ςτ2/)(α? - 2/). D

102 Chapter II D Repeatedly using Lemma 14 gives: η factors nStn(a)) = (αχ Λ V an) (аг Η h an) n —2 factors >(2αι 4- a3 h an)(2a2 4- a3 Η h an) (αϊ Η l· an) (αϊ Η l· an) > >nai(2a2 + a3 Η h an)(a2 4- 2a3 Η h an) (a2-\ h an_i 4- 2an) η factors D > > (ηαι)... (nan) = nn6n(a)n. The case of equality follows from that of the lemma. [GuJha]. (xxxiv) Gaines 1967 The following result is well-known, [DIp.75]. Lemma 15 [Schur] If λ^,Ι < г < n, are the eigenvalues of the complex matrix A = (α^)ι<*^<η then Y%=1 \\i\2 < E"j=i \αυ\2 with equality if and only if A* A = AA\ D If / 0 αϊ 0 0 0 α2 0 * 0 0 0 ··· αη_ι Van 0 0 ··· 0 / the eigenvalues are all equal to Φη(α) and so by Lemma 15, Σ™=1 а2 > п<Зп(а2), which is equivalent to (GA). D [Gaines]. (xxxv) О'Shea 1968 О'Shea gives the following lemma. Lemma 16 Let α be a non-constant decreasing η-tuple and if 1 < m < η define an+m = an, 1 < m <n. Then η m (23) further if 7П > 1 this inequality is strict. D We first remark that if m < η the last term of the sum in (23) is negative. Further if any term (23) is negative so is the succeeding term.

Means and Their Inequalities 103 To see this suppose that for some io < η and m < η, a™ — Y\T=1 aiQ+j < 0. Then obviously io -{- m > n, but also since ax > · · · > an, ax > an we must have that zo 4- m < 2n. This implies that aj0+m+1 = ai0+m+1_n. Hence since n>z0 + l>z0 + l-fm — n, Пт Пга — rL· J-· So ra ra a™+1 < a™ < ]^[aio+j < Д aio+1+j, j=i j = i or ra <+ι-Πα^ + 1+^'<0· The proof of the lemma is by induction on m and it is obvious that (23) holds with equality if m = 1. Let r-th term be the first negative term when we can write (23) as r—i m η m Σ (a™ - J]ai+j) >Σ(Ι[ ai+> - aT)■ (24) i=i j=i i=r j=i where all the terms in both sums are non-negative. Note that ar < di,i < r, ar > αι,ί > r with at least one of these inequalities being strict. Suppose that (24) holds for some ra, 1 < m < n, and multiply this inequality by ar to get, using the above remark, r — 1 m η m Σai (< - Π а*+з) > Σai(Π ai+> - <)> i=i j — i i—r j—i which is just (23) with ra replaced by (m-f 1). This completes the proof of the lemma. D D To prove (GA) using Lemma 16 note that is just the case m = η of that lemma. Π [O'Shea]. (XXXVI) DlNGHAS 1968 In various papers Dinghas has obtained several identities from which (GA) follows. Some are listed below; for proofs the reader is referred to the original papers. (α) η(21η(α) - βη(α)) = £ (α,·Λ - ©г(а1Л)) Ρί-^ϊ*. β<-ι(αιΛ)),

104 Chapter II where P0(x, y) = 1, Р^{х, у) = £}=2(j - \)х*-*у*~\ 3 < г < п. (b) Я£(а) - &пп(а) = £ ίΠ^αΛ ^ - ^-^Рг^Ща),^^)), where Pi-2 is as in (a). (c) ok f \ = exp ( Σ ™^'(α* ~ а^)2-^(аг,^-,21п(а;^))У ,οο χ where Wn = 1, and F{x, y, z) = ^ (χ + u){y + ^ + м) du. (rf) n^~; =ехрГ^ги»(а» - 2ln(a;w))2 J(a*,2ln(a; w))), where Wn = 1 and J(x, y) = I jz w ттт diz. V Уо (1 + и){х + у + и)2 [Dinghas, 1943/44, 1948, 1953, 1962/63, 1963, 1966, 1968). (xxxvii) Mitrinovic h Vasic 1968 This is a very simple proof of 2.3 Lemma 8(a) that can easily be modified to give a direct proof of the general weight case. D Let α be as in 2.3 Lemma 8(a), then by right-hand inequality in I 2.2(9), η η η η y^ log di - ^2 αΐ+n - °' °r Σαί -n ^ ^og (Πai)= n' г=1 г=1 г=1 г=1 D [Mitrinovic & Vasic 1968, p.27). 2.4.5 Proofs Published Between 1971 and 1988 Proofs (xxxviii)-(lxii) (xxxviii) Yuzakov 1971 D Let α be a non-constant η-tuple with A = 2ln(a), and Xi = a^ — A, 1 < г < η, when Х]^=1Жг = 0. Assume without loss in generality that X2 < 0 < x\\ then αια2 = (A 4- x\){A 4- #2) < ^(^ 4- xi 4- #2)· Hence, assuming (GA) for positive integers less than n, /,Al , ч xi/(n-i) . (A4-X1 4-Χ2)4-α3Η 4-αη А4-д?1+ж2аз...ап < = A 4 7 η — 1

Means and Their Inequalities 105 which, on using the above inequality /αια2 \ΐ/(η-ι) A> (——α3...αη) which completes the proof of (GA). D [Yiizakov]. Remark (i) Quantities similar to the χ>ϊ are used in 2.4.3 proof (xxii). (xxxix) Segre See VI 4.5 Example (Hi). [Segre]. (xl) Chong Κ Μ 1976 This is based on a result of Chong Κ Μ I 3.3 Corollary 17. rin ; D Assume that Д ai = 1 and put a^ = τ-τη"7-1 = — , 1 < J < η — 1, and i=l llj=i+iaj ci an d an = γψη — —· Then d is a rearrangement of c, and so by the I 3.3 Corollary 17, i=i г=1 ^ [CJhongKM1976c]. (жй) Chong Κ Μ 1976 This proof is based on I 3.3 Theorem 18, and the following lemma of Chong Κ Μ. Lemma 17 If a is an η-tuple and A = (2ln (a), 2ln (a),..., 2ln(a)) then A -< a. D Assume, without loss in generality, that α is decreasing, when if 1 < к < η—I к EIU+i ^ < (η - /с) £*_х <Ь, or fc2tn(a) < Σα*' * - fe - n ~ L Since of course n2ln(a) = 2^=1 a^, we get that A -< a. D D Now by Lemma 17 and I 3.3 Theorem 18 η η which is just (GA). Π [Chong К М 1976с, 1979). (xlii) Myers 1976 This is based on the idea in 2.4.2 proof (ж).

106 Chapter II Lemma 18 If x_ is a real η-tuple with Σ™=1 x% = 0 then 1 > Πϋ=ι(1 + x^ w^n equality if and only ifx_ is constant. D If η = 2 then X2 = —x\ and so (1 4- #i)(l 4- £2) = 1 — x? < 1, with equality if and only if xi = Ж2. Now assume the lemma has been proved for all integers less than η, η > 2 and that χ is not constant. Then not all a^, 1 < г < η, are zero, and assume without loss in generality that xn-i <0 < xn. Then, П Π—2 П(1 + ж») <(1+жп-1 + жп) JJ(1 4- ж»), since χη-ιχη < 0, <1, by the induction hypothesis. This completes the proof of the lemma. D D Now let α be a non-constant η-tuple, with b defined as in 2.4.2 proof (ж), that is di = (1 4- Ьг)21п(а), 1 < г < η, when, as in the earlier proof, Σ™=1 h = 0, and not all of the terms of b are zero. Then 0n(a)=0n(l4-b)Sln(a) <2tn(a), by Lemma 18. D [Myers]. Remark (ii) A simple proof Lemma 18 can be based on I 2.2(8). The same η η method will prove: У^ХгУг — 0 => 1 > TT(^ "^ Ж*)У% ^'£/ι equality if and only i=i i=i if χ is constant. Chong Κ Μ 1976 D Let α be a non-constant η-tuple and assume without loss in generality that a\ = mina < an = max a. Further assume that (GA) has been proved for integers less than n. Then КШ =Sln(a)St^i (αϊ 4-αη - On(a),a2,.. .,Οη-ι), >2ln(a) (a\ 4- an — 2ln(&))a2 .. . an_i, by the induction hypothesis, = (aian 4- (an - 2tn(a))(^n(a) ~ «i))«2 ·· · an-i > «ia2 .. .an, which is (GA). D [Chong· Κ Μ 1976b). Remark (iii) Chong has given a similar proof using 0n(a); [Chong Κ Μ 1976a].

Means and Their Inequalities 107 (xliv) Bellman 1976 Bellman has given a proof of (GA) based on developing identities for 2l2fc (&) ~ <32k(a) and the use of 2.2.4 Remark (1). D First note that a\ + a\ = 2axa2 + (αϊ - α2)2, (25) and so a\ + a\ + aj 4- a\ = 2{αγα2 + аза4) 4- (αϊ - α2)2 4- (α3 - α4)2. (26) Substituting α^ for αϊ, α2 for α2 etc. in (26) and using (25) we get a\ 4- a2 4- a3 4- a\ = 4aia2a3a4 4- 2(aia2 - a3a4)2 4- (a2 - a2)2 4- (a2 - a|)2. (27) Now add (27) to a similar identity using а5,аб,а7, ag and replace a\ by a\ etc, to obtain a similar identity in eight variable with a\ 4- · · · 4-a§ on the left, 8aia2 ... ag the first term on the right, and all the other terms on the right being non-negative. In this way we obtain an identity of the form 2k 2k У ai =2 1 I (Ц 4- non-negative terms, from which (GA) for η-tuples with η = 2fc, /c = 1,2,... is immediate. D [Bellman 1976]. (xlv) SCHAUMBERGER h KABAK 1977 This is a particularly simple inductive proof. n+i D Clearly ]T (a? - α^)(^ - αά) > 0. So, i,j = l П+1 П+1 П+1 «Σ«"+1^(Σ*Σ«»?) J** П+1 П+1 : ( 2_^ ain \\ aj) > by the induction hypothesis, n+i =n(n 4-1) Π aJ· J=i This gives (GA) and the case of equality is easily obtained. D [Schaumberger L· Kabak 1977].

108 Chapter II (xlvi) 197712 D We may suppose without loss in generality that βη(α;υ;) = e, and Wn = 1. By I 2.2(7) we have a^ > eloga;, 1 < г < η, so η η 21п(«;ш) = Σ^α* > elog ( Д «Г) = е = ^nfew). The case of equality is easily obtained. D (xlvii) Climescu 197713 D Assume, as we may, that the a is an (n-f- l)-tuple with nlLi* ai = 1. If then we have proved (GA) for the integer η we have that αϊ + · ■ · + αη > η(αι... αη)1/η (28) Adding over n similar inequalities we obtain n+i n+i v -, /_ n+i kl/n αϊ + ···+αη+1>^(Π^) η-Σ^1/η· (29) Using the right-hand side of (35) as the left-hand side of (34) and repeating the argument, and then iterating this we arrive after r steps at n+i αϊ Η h αη+1 > Σ α* r/n. Letting r —► oo gives αϊ Η h αη+ι > η 4-1, which by 2.3 Lemma 8(b) completes the proof. D Remark (iv) Of course this proof does not give the case of equality. (xlviii) Mijalkovic 1977 14 D Assume that 0n(a) = 1, and that (GA) is known for positive integers less than n. 2C„(a)=^ + ^^Qln_1(a) η η >— Η Φη-ι(α), by the induction hypothesis, η η =^ + ^a;1/"-1 > 1, by 2.2.2 (5), (η = 2 (GA)). D η η 12 I have no source for this proof other than [Ml\ 13 I have no proper reference for this proof 14 This proof is unpublished.

Means and Their Inequalities 109 (xlix) SCHAUMBERGER 1978 The following is a very simple calculus proof. D Put χ = an and f(x) = n0n(ai,..., an_i, x; w_) — wnx, and assume (GA) for integers less than n. Then / has a unique maximum at χ = xq = (*5η_ι(α; w) and as a result f{x) < f(xo). This inequality is Wn-i®n-i(a;w) > Wn&n(a]w) — wnan, from which (GA) follows by the induction hypothesis. D [Anderson D 1979a; Schaumberger 1978, 1990a]. (0 Landsberg 1978 The following interesting proof is based on the laws of thermodynamics. There was some criticism of this method of proving mathematical theorems, criticism that was itself objected to and the method extended; see the references and III 3.1.1 Theorem 1 proof (x) D Let a,i, 1 < г < η, be the temperatures of η identical heat reservoirs each having heat capacity c. Put the reservoirs in contact with each other and let them come to an equilibrium temperature A, say. The first law of thermodynamics, conservation of energy, tells us that A = 2ln(&)· The second law of thermodynamics implies a gain of entropy, that is to say en log(Α/<δη(α)) > 0. This implies that A = 2ln(a) > 0п(й); further there is equality if and only if there is zero entropy gain, if and only if all the initial temperatures are the same. D [Abriata; Deakin L· Troup; Landsberg 1978,1980a,b,1985; Sidhu]. (li) Zemgalis 1979 This is a simple inductive proof of (GA) in the form of 2.3 Lemma 8(a). D Assume (GA) has been proved for all integers /с, 1 < к < η, and that Πΐ=ι аг = 1 where without loss in generality we can assume an = mina, an+i = max a, when an < 1 < an+i. Then anan+i -f- 1 < an -f an+i, see 2.1 Lemma 3 proof(w). Now: η 4-1 =пбп+1(а) 4-1 = η(δη(αι,..., αη_ι, αηαη+ι) 4-1 <(αι 4- · · ■ 4- αη-ι 4- αηαη+ι) 4-1, by the induction hypothesis, <αι 4- · · · 4- αη+ι; by the above remark. The case of equality is immediate. D [Herman, Kucera & Simsa pp.ЦЗ-144, 151], [Zemgalis]. (Hi) Bullen 1979

no Chapter II The proof (ix) of 2.2.2 Lemma 6, the η = 2, general weight case, can be extended to give a geometrical inductive proof along the lines of a proof of (J), I 4.2 Theorem 12 proof (Hi). The induction is obvious from the following proof of the η = 3 case, where the notation is that used in the quoted proofs of earlier theorems. D Consider the function D3(s,t) =Vl3(x,y,z;l -s-t,s,t)- 03(х,2/,г;1 -s-t,s,t), definedonthetriangleT = {(s,£); 0 < s < 1, 0 < ί < 1, 0 < s + t < 1}. The proof of (GA) in this case consists in showing that except at the corners of Γ, D3 > 0, unless χ = у = z. We can assume, without loss in generality, that 0 < χ < у < z\ further, we can assume that 0 < χ < у < ζ; for if any two of x,y,z are equal then D3 reduces to a case η = 2 of (GA) and is known to be positive unless the remaining two are equal. Thus if у = ζ, and we put и = s 4-1 D3(s,t) = D(u) =Щх,у]1 - u,u) - &(x,y;l - u,u), 0 < и < 1; (30) and D(u) > 0, и φ 0,1, see proof (ix) of 2.2.2 Lemma 6. Being continuous the minimum value of D3 is attained somewhere on Г and if it is at an interior point then it is at a point satisfying -~ =(y - x) - ©3 bg(j//x) = 0, ^ =(*-*)-«3 log(z/a:)=0. That is at some point (sch^o) such that у — χ ζ — χ 1 ; = ι ] = 03(ж, y,z]l-s0- t0, s0, t0). (31) log у — log χ log ζ — log χ However the logarithmic function is strictly concave so since у < ζ the first ratio in (31) is strictly less than the second15, I 4.1 Lemma 2; so there is no such point (so, to), the minimum of D3 must occur on the boundary of T. However on a side of T, D3 is of the form D in (30), and is then positive except at the end points of the side. Hence D3(s,t) > 0 except at the corners of T. In the general case suppose that we have (GA) for positive integers less than n. If t = (ti,... ,in-i), Ш = (1 - Σ™=ι *>ί)> and Αι(έ) = %п(аж) - 0nfem), defined on the simplex Τ = {£; 0 < w. ^ 1}· Then as before we can assume that α is a strictly increasing η-tuple, for otherwise we are reduced to the case η — 1, and as The quantities in the two ratios in (37) are the logarithmic means £(x,y), £(x,z); see 5.5, VI 2.1.1.

Means and Their Inequalities 111 before Dn must attain its minimum on a face of Τ where it is positive except at the corners by the induction hypothesis. D [Bullen 1979, 1980]. (liii) Cusmariu 1981 In trying to extend the geometric proof (v) of 2.2.1 Lemma 3 Cusmariu obtained the following proof of (GA). D Consider the following nxn matrices: i" the unit matrix, J the matrix with entries all 1, and S the matrix whose elements α^+ι = 1,1 < г < η — 1, αη,ι = 1, and the rest are zero. If now V = (I + S)/2 = (vij)i<i,j<n, then V is doubly stochastic and so limm^00 Vm = — J. Alternatively this can be obtained directly. Now if α is a non-constant η-tuple and for m £ N let V™ = (v\j)1<ij<n and define , η η (m) i=i j=i In particular /0 = 21п(д) and lim™^^ fm = <6n(a). Further fm=l(f(a,V™) + f(a,SV™)) >Vf^Vm)f(a,SVm), by 2.2.1 (2),((GA)with η = 2 and equal weights), =f(a,Vm+1) = fm+i. These facts imply (GA). D [Cusmariu]. (Iw) VAN DER HOEK 1981 This is a simple inductive proof of 2.3 Lemma 8(a). D Assume that Y[^=^ α* — 1 and put s = aJ/Д when ΠΓ=ι(δα*) ~ 1 an<^ so ^У the induction hypothesis Y^=1(sdi) > n. Hence, by I 1.2(6), n+i 1 η Y^ai = -Y^(sai) + sn>-+sn>n + l. D i=i i=i [van der Hoek]. (Iv) Chong KM 1981 D Let α an η-tuple with αλ < an < an_x < · · · < a3 < a2, αλ =£ a2, and for some r, 1 < r < η — 1, define и — α^ηα^7 ' v = аГ^Паг+1 · · · α\ίη-> x = /α ι \ 1/n ( r J . Then η > -у and applying I 2.2 (22) we get that . г-, ^ 1/n (n — l)/n . г-, ar+i + Pr+i > a/ aj.+1 " + Pr,

112 Chapter II where Pr = ajnaj^ ... a\[n', 1 < r < η — 1, and An = ax; this inequality is strict when r = 1. Sum these inequalities over 1 < г < η — 1, to get nSlnfe) >6n(a) +^a;/n4n_1 )/n 71 -i // - >0n(a) + (n - 1) ( Y[ α\/ηα^~ι)/ηλ , by the induction hypothesis, =n<5n(a). This gives (GA). D [Chong Κ Μ 1981]. (Ivi) Ruthing 1982 Ruthing has given several inductive proofs using the symmetric form of (B), the inequalities I 2.1(2). D All the proofs use the inductive hypothesis, 0n-i < 2ln-i, and an inequality obtained from one or other of the inequalities I 2.1(2) by substituting a = η and various expressions for a and b. Thus using the left inequality with: (a) a = α^(η-1}, b = ^/(η~1}(α), gives аЖп~А < K\ (b) a = aj/(n-1}, b = <3n/(n-1)(a), gives <3n < i((n - 1)0η_ι + an); and using the right inequality with: (с) а — 21^_!(а), Ь = аУn, gives the inequality in (a); (d) a = 2ln_i, Ь = 2ln, also gives the inequality in (a); (e) α = 6η(!\(α), Ь = αη , gives the inequality in (b); (f) a = 0n_i(a), Ь = Φη(α) also gives the inequality in (b). D [Ruthing 1982]. (Ivii) Wellstein 1982 Taking /(α) = ΠΓ=ι a* m * ^.2 Theorem 13 gives the equal weight case of (GA). [Wellstein] (Ivill) SCHAUMBERGER 1985 D Assume that Wn = 1. Then by the right-hand inequality of I 2.2(9), ^i>Wi + Wi ^(сц/вп) = ил + log(af/C), 1 < i < n. Summing over г gives ^>^η + 1οδ((5η/((5η) = 1. The case of equality is immediate. D [Schaumberger 1985a, 1988].

Means and Their Inequalities 113 (lix) Soloviov 1986 D By I 4.6 Example (viii) χ (α) = ^n(a;w) is strictly concave on the cone (R+)*, and, assuming that Wn = 1, Vx(e) = w. (GA) now follows by an application of the support inequality, I 4.6 (~24), with U. — <L-> 21 = й- П [Soloviov]. (Ix) Teng 1987 This proof of the equal weight case depends on the following lemma that is another generalization of I 2.2(20). Lemma 19 The equal weight case of (GA) is equivalent to the inequality η 1 (32) with equality if and only if а = е. D Given (GA) the left-hand side of (32) is greater than or equal to / η \ 1/(η + ι) there is equality if and only if ax = · · · = an = =^ , that is, if and only if IL=i ai a-L = - · · = an = 1. Now given (32) and a positive real a we have that n-i Σαί , 1 ^ that is n-i 1 Σα* + αηΗ^~~ -αη· i=i IL=i α* Taking a = (ΠΓ=ι агУ^п m ^ne ^as^ inequality gives (GA). The case of equality is immediate. D D As a result of Lemma 19 to prove (GA) we need to prove (32),which we do using a short version of Teng's proof due to Hering; [Hering 1990]. The case η = 1 is just I 2.2(20) so suppose (32) holds for η, η > 1. Then n+i ^ η ^ ι ^ > n+i +1 + (on+1 - i) (i - -4i—). (33)

114 Chapter II by the induction hypothesis. Now either Π7=ι α* — 1 when at least one ai > 1, or ΠΓ=ι аг < 1 when at least one a^ < 1. In either case we can, without loss in generality, assume that this a^ is an+i- Hence the last term on the right-hand side of (33) is non-negative and the proof of (32) is completed. D [Teng]. (Ixi) Minassian 1988 This proof of the equal weight case is a mixture of induction and the use of calculus. D Let n>2,a= (αϊ,... , αη_ι,χ), s = αχ -f «2 Η f-an_i, ρ = а\аъ ... αη_ι and define /(ж) = nn(2l£(a) - 0£(а)),ж > °· Then f(x) = (x + s)n - nnpx, f(x) = n(x + s)n_1 - n>, f"{x) = n(n - l)(x + s)n"2. Hence / is seen to have a minimum at χ = xq = np1^71-1) — s, and simple calculations give that f(xo) — nnp[s — (n — l);?1^71-1)). Now the induction hypothesis is s > (n — l)^1/^-1) with equality if and only if a\ = · · · = αη_ι— a property that clearly holds when n=2, 3. So by the induction hypothesis f(xo) > 0, with equality if and only if χ = x0. D [Minassian]. Remark (v) Minassian points out that this argument allows for consideration of certain real α if (GA) is written in the form 21^(а;ги) > 0™(а;ги)· (Mi) Yu 1988 D In the right-hand inequality of I 2.2 (9) put χ = ai/<6n(a;w_), and multiply by Wi, 1 < i < n, and assume that Wn = 1. This gives Wi log α^/Φη (α; w) < u^ — Wi, 1 < г < η. Adding over г gives (GA), and the inequality is strict unless α is constant by I 2.2 (9). D [Yu]. Remark (vi) This proof is similar to proof (Iviii). 2.4.6 Proofs Published After 1988. Proofs (Ixiii)-(lxxiv) (IxUi) SCHAUMBERGER 1989 D Put χ = aie/<3n(a), 1 < г < η, in I 2.2 (7) and multiply to get eV βη(α) ) >(еП^=1аЛе = епе On simplifying this gives the equal weight case of (GA). The case of equality is immediate from that of I 2.2 (7) . D [Schaumberger 1989].

Means and Their Inequalities 115 (Ixiv) Ben-Tal, Charnes h Teboulle 1989. See VI 4.6 Remark(vii). [Ben-Tal, Charnes к Teboulle]. (Ixv) Alzer 1991. See 5.5 where an interesting proof is given using Cebisev's inequality. [Alzer 1991a]. (Ixvi) Schaumberger h Bencze 1993 Assume that we have the following inequality: if χ is an increasing non-negative η-tuple then Xn > #i(2£2 -£i)(3x3 - 2ж2)' * · (nxn -n- lxn-i), (34) with equality if and only if χ is constant. Now, in this inequality, put α\= Χγ,α^ — 2ж2 — #ъ «з = Зжз — 2ж2 ... to get the equal weight case of (GA), together with the case of equality. To prove inequality (34) note that it is trivial if η = 1 and assume that it is valid for some η > 1; then xi(2x2 - Ж1)(3жз - 2ж2) ■ · -(nxn - η - 1жп_1)(п + lxn+i - nxn) <x^{n + lxn+i — гсжп), by the induction hypothesis, =<+1(^H:i=±i -n) < хЦ\, by I 1.2(7). The case of equality follows from that of I 1.2(7). An alternative proof of inequality (34) is given in 3.7 Example (i). [Schaumberger L· Bencze 1993]. (Ixvii) Alzer 1996 D Assume Wn = 1 and that α is an increasing η-tuple. Then there is a /c, 1 < к < η, such that а^ < <бп{а',ш) < α&+ι· So £g -1 - |>/>■ - o<<+|>/;w -«-)- Since both sums on the right-hand side contain only non-negative terms we get that the left-hand side is non-negative, which is (GA). Further the right-hand side is zero if and only if a^ = (Sn, 1 < i < n, which gives the case of equality. D [Alzer 1996a].

116 Chapter II (Ixviii) Lucht 1995 D Let a,w_ be η-tuples, and assume that Wn = 1 and a is not constant; let χ > 0, then by the right-hand inequality of I 2.2 (9) we have for each г, 1 < г < η, di αϊ Wi Wi> Wilog—; X X further this inequality is strict unless ai/x = 1. Adding and noting that at least one αϊ/χ φ 1, we get о* , \ л Фп(а;ш) ™n (α; Ш) > χ + χ l°g · Taking ж = (6n(a;w) gives (GA). [Lucht]. Remark (i) This should be compared to 2.4.5 proofs (Iviii), (Ixii). (Ixix) Usakov 1995 In an interesting paper Usakov developed a unified method of proving inequalities based on the following simple lemma. Lemma 20 Suppose that Μ С R and F : Μ ь^ R, where F(x) = Σ"=1 ft{x), then η η V inf fi{x) < infF(x), V sup ft(ж) > sup F(x) ^rcGM хем ΖΞίχΕΜ xeM with equality in either inequality if and only if all the extrema occur at the same value of x. To prove (GA) assume let α and w_ be η-tuples with Wn = 1. Let ft(x) = Wi(ii(x — log ж), 1 < ί < п,ж Ε Μ = R+ then ft has a minimum at Xi = Ι/α*, 1 < г < n, giving £?=1 infxGM ft (x) = 14- log(0n(a; w)). F(x) = %1п(а',ш)х — log ж which has a minimum at ж = 1/21п(а;ш) giving inLceM F(x) = 1 4-log(2ln(a;u;)). Using the lemma we get (GA) together with the case of equality. [Sandor L· Szabo; Pecaric L· Varosonec; Usakov]. (hex) Hrimic 2000 D Let α be an η-tuple and define the η-tuple b by, b\ = —, b2 = —s ,... ..., bn = — " ' = 1. A simple application of I 3.3 Corollary 17 gives, ^ h 1 , h 1 , , h 1 αι , a2 , an Лг bi bn_i βη(α) 0η(α) 0n(a)' from which (GA) is immediate. D

Means and Their Inequalities 117 [Hrimic]. (IxXl) SCHAUMBERGER 2000 This is another proof that uses the the right-hand inequality I 2.2 (9); see also 2.4.4 proof (xxxvii), 2.4.5 proofs (Iviii), (Ixii) a,nd(lxviii). D η > V^ log ( ) ), by the the right-hand inequality I 2.2 (9), ~ \On[a)J "ь«П(г!1г)-а D [Schaumberger 2000]. (Ixxii) Gao Ρ 2001 D Assume without loss in generality that Wn = 1 and α is an increasing n- i terms tuple with αχ =fi an and write x_{ = (ж,..., χ, a,i+u ... an) and define the functions Di(x) = 2ln(^;u>) - ®n(Xi',W.), ,0 < χ < αΐ+ι, 1 < г < п. Then D[{x) = Wi(l- 6n^;-M < o, by strict internality, 1.2 Theorem 6. Hence Di is strictly decreasing and so Α(αι) > АЮ = Α(α2) > Α(α8) = Α(α3) > · · ■ > Dn{an) = 0, with at least one of these inequalities being strict; but D1(a1) > 0 is just (GA). D [GaoP]. ^ (Ixxiii) Rooin 2001 D This proof proves the equal weight case of (GA) for rational η-tuples a. It is then easy to see that we can assume this η-tuple consists of integers and even that 2tn(a) = nk where к G N*, к > 2. Now the set A = {α; α Ε (N*)n and Sin (a) = nk}, for a given positive integer к is finite. Hence A contains an a/ for which the product of its elements is maximum; that is η η

118 Chapter II Suppose that gf is not constant then there exist two elements of a/, without loss in generality ax/ and a!,, such that a[ < к < α'2, which implies that a'2 — a[ > 2, and a'2 — a[ — 1 > 0 . Now a" = (a[ -f 1, a!2 — 1, a'3,..., aJJ G A and η Π α? =K + l)(a'a - 1)α'3... <4 = {а[а'2 + а'2 - а[ - 1)а'3 ...а'п i=i η >ικ г=1 This is a contradiction and so α/ is constant. Hence a[ = · · · = a!n — к and this implies (GA). D Remark (ii) It is not immediate how to use this result to get (GA) for all positive n-tuples. [Room 2001c]. (Ixxiv) Hasto 2002 This proof by induction is an elaboration of 2.2.1 Theorem 3 proof (ж), in the form suggested there in Remark (vi), and is related to Liouville's proof, 2.4.1 proof (Hi). D Assume without loss in generality than a an η-tuple such that a\ < · · · < an. Let χ = an and put f(x) = %ln(a;w_) - ^nfe^) when Hence / is increasing if χ > 0n-ifew), in particular by the internality of the geometric mean if ж > an-x. Hence f(x) > f(an-1) = 2tn-i(ani Ш*) ~ Φη-ι(^ή; Ш.') where w_ is the η — 1-tuple (wi,..., wn-2, wn-i 4- wn). The result follows from the induction hypothesis. D [Hasto]. Remark (iii) As in the proof in 2.2.1 this gives a little more as it shows 2ln (a; w) — ®n (й.; ш) is increasing as a function of any term exceeding the geometric mean of the remaining terms, and is decreasing as function of any term that is less than the geometric mean of the remaining terms. A similar proof, that also gives a little more, can be based on the ratio 21п(^;ш)/Фп(а;:ш)· Remark (iv) Reference should be made to a slightly different use of the function / in proof (iii) of 3 .1 Theorem 1. 2.4.7 Proofs Published In Journals Not Available to the Author These are listed in chronological order.

Means and Their Inequalities 119 [Schlomilch 1858a, 1859], see III 3.1.1 Remark (iv); [Unferdinger 1867; Tait 1867/69; Schaumberger 1971, 1973, 1975a, 1985b, 1987, 1990b, 1991, 1995; Moldenhauer; Pelczynski 1992]. 2.5 Applications of the Geometric Mean-Arithmetic mean Inequality We have already seen a simple but effective use of (GA) in Heron's method, 1.3.5 above. Another application was given by Kepler. He noted that an ellipse with semi-axes α and b has the same area as a circle with radius y/ob. Hence from the classical isoperimetric property the perimeter of the ellipse is bigger than the circumference of that circle, 2π\/α6. Using (GA) Kepler gave π (α 4- b) as an approximation for the perimeter of the ellipse16. We give some further uses below; of course there are many more, see for instance [Monsky; Ness 1967). Example (i) Determine all the polynomials xn-\-an-1xn~1-\ hao, with a\ = dbl, 0 < г < η — 1, having all real zeros. D Suppose that the zeros are Xi, 1 < г < η, then Σ™=1 x^ = αη-ι ~ 2αη_2 = 1 zb 2 = 3, since the sum must be positive; further ΠΓ=ι xi=ao = 1· Βν (GA) 1 < 3/n, so η < 3. It is now easy to look at the finite number of possible polynomials particularly since when η = 3 the equality case of (GA) implies that the zeros are all ±1, which gives αχ = — 1; [Boyd]. D Example (ii) The proof of a surprising result due to Simons, [Simons], is based on (GA). Let a be an η-tuple and α a real η-tuple all of whose terms are distinct and define f{x) = ^гп=1а^, χ > 0. If д(х) = аха is chosen to give a good approximation to f at the point χ = x0 in the sense that f(xo) = g(xo) and f'(xo) = gf(xo) then f(x) > g(x) with equality if and only if χ = x0. D The conditions imply that а = Y™=1 сцоцх^1 /f{xo) and а = f(xo)xQa- Hence f{x) = srn η ^i /(zo)>M( —) Джо), by (GA) l^i=i aiX0 i=1KX0y ( X \ a = ( —) Джо) = аха = g{x). There is equality if and only if ((ж/ж0)а% 1 < г < η) is constant; that is if and only if χ = xq. D 2.5.1 Calculus Problems Problems solved by elementary calculus can often be solved using (GA), usually in its simplest forms 2.2.1(2) or 2.2.2(5); [Niven], [Boas &; Klamkin; Frame; Lim]. The classical isoperimetric property is: of all convex closed curves of a given area the one with the least perimeter is the circle. It is known that the perimeter of an ellipse cannot be expressed in terms of elementary functions; [CE p.521; EM5 pp.206-207].

120 Chapter II Example (i) Ншп_юо -γ/η = 1; [Room 2001a]. Assume that η > 3, then: 0 < Jfii - 1 = . n~l ~——j- < П~1 , by (GA), n- 1 1 < n(3n-l)/2n - ffi Alternatively, [Sinnadurai 1961): η — η terms 1 < η1/2" =(η«/2)1/"2 < 1 + -·· + 1+η^+-·· + ^, by (GA), _П2 - П + Пу/П _ 1 _ 1 , J_ . ι , JL This gives: 1 < n1'*1 = in1/2"1)2 < 1 + A=- y/n 2.5.2 Population Mathematics An application to population mathematics, [Anderson D 1979b], is based on estimating the lower bound for the positive root of the equation wi 4- W2X 4- · ■ · wnxn~1 — xk = 0, (35) where υ; is positive and /c > η — 1 > 1. Using the results of I 1.1 it is easily seen that this equation has exactly one positive root which we will assume is not equal to 1, for in that case the discussion is trivial. Equation (35) can be written as 0 = —- 2_^wiXl x — —-, using (GA) we easily 1=1 , n ν 1/Wn xk get that 0 > ( ТТж^_1^г" J — ——. This leads to a lower bound for ж, namely i=i 1 n x > W%, where а = к — —- 2^ Wi+i- Wn t=i For further discussion the reader is referred to Anderson's paper. 2.5.3 Proving Other Inequalities This is of course one of the main applications of (GA), and much of the rest of this book is evidence of this. Here we give a few isolated examples of this use, many more can be found in [Herman, Kucera & Sims a pp.151-166]. (a) It is possible to compare the geometric and arithmetic means with different weights; [Bullen 1967; Dragomir L· Goh 1997a; Iwamoto; Mitrinovic L· Vasic 1966a,c; Wang С L 1979d].

Means and Their Inequalities 121 Theorem 21 If a,u,v_ are η-tuples then: (a) with equality if and only if avu-1 is constant; (b) K®n(a;u); \®n(a;v); \®η(α',1Α+ν) D (a) It is easily seen that which implies the result by (GA). (b) This is an immediate consequence of the similar inequality associated with (J), I 4.2 Theorem 15(b), applied to the negative log function. D (β) The following theorem is in [Lupa§ Sz Mitrovic]. Theorem 22 If α is an η-tuple then 1 + 0η(α) <&n(e + o) <l + 2ln(a); (36) 1 + ^—гт <0n(e + a"1) < 1 + 6n(a) ~ ~ Йп(а)' There is equality if and only if α is constant. D By 1.2(7) and 1.2(10) it suffices to prove (36). The right hand inequality is an immediate consequence of (GA) and the observation that 2tn(e + a) = 1 + Sin (a). For the left inequality consider, η η Π(ι+α*)=ι+Σ)Σ!(β<ι···ο i=i m=i m ^+Е£)П!К'-^иу(СА), ra=i ^ ' m =1+е(;)(п^/"=(1^-й)" 771=1 ^ ' j = l The case of equality follows from that of (GA).. D Remark (i) Another proof of (36) has been given by Keckic, [Keckic].

122 Chapter II Remark (ii) A part of this inequality has been generalized by Pecaric, see I 2.1 Remark(v), and similar but more general results can be found in some papers of Kovacec and others; [Alzer 1990o; Kovacec 1981a,b; Mitrovic 1973; Pecaric 1983a]. In particular it is immediate from the above proof that the right-hand inequality in (36) holds for weighted means. See also III 3.1.3 Remark (iv). (7) The following inequality is in [Kalajdzic 1970]. Theorem 23 Let a,w be η-tuples with Wn = 1 and let b > 1, then: ^ ~ (n- 1)! ^ with equality if and only if α is constant. D Let b = №- = (bai,..., ban), w. = (ги^,..., Win) and consider the sum on the right-hand side of the above inequality Y^[b(3n(a-w) <^|^(α;ώ)5 by (QA^ = Y^n(kw) η <Ymn(b-w) = (n-i)\^2ba\ The case of equality is immediate. D (δ) (GA) can be used to give a simple proof of a special case of I 2.2(11); the case when ρ = η -f l,q = Щ [Forder; Georgakis; Goodman; Melzak; Sinnadurai 1961]. l/(n+l) (l(l + -)n) <-LT(l + (l + -) + ---+(l + -)Yby(GA), χ that is X\n /„ X \n+l (ι + -Γ<(ι + -^-)τ 4 η' ν η 4- 1 Remark (iii) This shows (l -\—) increases strictly with n, and a similar ar- 4 n/ gument can be used to show that (l Η ) strictly decreases with n; see I 2.2(a). (б) Here we collect various results, without proofs.

Means and Their Inequalities 123 Theorem 24 (a) [Mijalkovic L· Keller; Lyons] 2tn(a;w) < <3n(a;au>). (b) [Zaciu] /gin(a)\a"Ca) Sw(o&) (cj [Daykin & Schmeichel] Ifan+k = dk, bk = Ak+n - Ak, 1 < к < η, then η<δη(α) < 0П(Ь). (d) [ΑΙ, ρ. 348] lib?а > 0, 0 < j < к, and Afc+1a = 0 then /га + η — 1 χ %m+n(a) V fc Яп(й) /n-1 (е) [AI pp.208-209, 345-346], [Mitrinovic, Pecaric L· Volenec], [Klamkin 1968, 1976; Pecaric, Janic L· Klamkin 1981] If α is an η-tuple define the η-tuples b,c by bk = Sin (a) — (n — l)afc, and ck = , 1 < к < n; and assume that b is non-negative then η — 1 0n® < ®n(a) < ®n(0)· (f) [AI p.344], [Akerberg] 1 JL Ml/fc fe=i (g) [AI p.215], [Schaumberger 1975в] IfO < a< b then Remark (iv) The inequalities of (e) extend the results quoted from [AI]. Remark (v) The result in (g) has been extended; see below 4.1 Theorem 2. (φ) The inequality I 2.2 (24) is a generalization of I 2.2 (20); another generalization is the following; see [Janic L· Vasic].

124 Chapter II Theorem 25 If χ is an η-tuple and ifl<m<n, then Xl-\ l· Хш X2 Η l· £m+i Xn Η hXm-l ПШ ^m+1 Η l· Xn Xm+2 Η 1" Xn % Η l· Xn-1 Π - ΤΠ D The left-hand side of this inequality is equal to Xn ~ (жт+1 Η l· жп) Xn - (жт+2 Η hxi) Xrn+l Η l· Жп Жт+2 Η l· Xl ^η ~ (Xm H hXn-l) 2-m "г * " " "τ Χη — 1 1 ■ + ··■ + ■ П2Хп >- г г — П, (жт+1 Η l· Жп) Η l· [Xm Η l· Xn-1) by (HA), n2Xn nm η (η — m)Xn η — m' D 2.5.4 Probabilistic Applications The mean, or expected value, of a random variable X assuming the discrete value Xi with probability pi, 1 < г < η, is E(X) = Σ?=ιΡΐχί· Hence we can write Яп(&ш) = ЩХ); ®n(a;w) = exp(£(logX)); fin(a;w) = l/E{l/X). Then (GA) can be expressed as £7(log(l/Y)) > 0, where Υ = X/E(X); the inequality between the geometric and harmonic means can be written as £(log(l/Z)) > 0, where Ζ = (l/X)/E{l/X) and (HA) can be written as E{X)E{l/X) = E(l/Y) = E(l/Z) > 1. The further discussion of these results is outside the scope of this book and reference should be made to [Pearce }.

Means and Their Inequalities 125 3 Refinements of the Geometric Mean-Arithmetic Mean Inequality 3.1 The Inequalities of Rado and Popoviciu There are many refinements of (GA) and we first consider those that result from rewriting this inequality in one of the forms Ма;ш)-0п(а;ш) -° (*) ^nfe^) >1 (2) and then asking if the left-hand sides of (1) and (2) can be given more precise lower bounds, in general, or for η-tuples satisfying certain conditions. The consideration of this question often leads to still further proofs of (GA). Theorem 1 If"α, υ; are η-tuples, η > 2 then )). (R) )1/Wn / ч 1/Wn_i with equality in (R) if and only if an = Φη_ι(α; w), and in (P) if and only if an = Stn_i(a;u;)· D We give several proofs of these results. (г) First we consider the case of equal weights and use a method due to Jacobsthal, see 2.4.3 proof (xxi) of (GA); [Jacobsthal]. This is just (R), in the case of equal weights. Further, from I 1.2(7), there is equality if and only if 0n(a) = 6η_ι(α), or equivalently if and only if an = ^n-i(fl). Similarly, 21η(α)\η_721η-ι(α) 6η(α)/ ν^η-ι(α) 1 ι V ^ _ Λ\η**η-ι(β) η \ Sin (α) // αη Kfeis) ·*<->·

126 Chapter II The case of equality follows from that of (~B). (ii) Simple calculations show that (R) is equivalent to ™n Wn^1 . wn/WniiKWri^1/WTli ч ,Qs W~an + -^Г-Яп-АО^т > a-n ' ^n-i (а;ш), (3) which is a consequence of (GA). A similar reduction gives a proof of (P). (Hi) This is a modification of 2.4.1 proof (Hi) of (GA) and is due to Liouville; [Liouville]. Let χ = an and /(ж) = Wn (2ln (α; υ;) — Фп(а;ш))· Differentiation shows that for ж > 0, / has a single turning point, a minimum, at χ = x0 = ^n-ifew)- Simple calculations show that /(жо) = Wn-i(2ln-i(a;w) — ^n-ifeu;))· Hence/(ж) > f(xo), x φ xoj and this completes the proof of (R). If instead we consider я(ж) = f J1 ~' ~~4 ) a similar discussion leads to (P). V<3n(a;w)/ v y (w) (R) is an immediate consequence of I 4.2(7) in the case f(x) = ex and the сц there are replaced by loga^, 1 < г < п. In a similar way we get (P) by putting f(x) = —logж. (ν) The following ingenious proof is due to Nanjundiah who obtains (R) and (P) simultaneously by appealing twice to (~B) in the form I 2.1(3); [Nanjundiah 1946]. * > %п(а;ш) 9Ln-i(a;w) = an wn wn *nVtem) ^(а;ш) Ч1' > вп(а;ш) Фп-А&Ш)· wn wn (vi) A simple proof based on (B) has been given recently; [Redheffer Sz Voigt]. Rewrite (3) as an > &п(а;ш) 6n-ifel, (4) wn wn and notice that this is just (~B) with 1 4- χ = &п(а]ш)/®п-1(а',ш) and a = Wn/wn. (vii) Proofs are easily obtained from the inequalities in (a) and (b) of 2.4.5 proof (M)\ [Ruthing 1982]. D Remark (i) Repeated application of (R) or (P) leads to (GA) together with the case of equality; also repeated application of (R) gives a particular case of I 4.2(7); Wn(Vln(a;w) - <3n(a;w)) > Wn-i(%b-i{&w) - <6n-i(Q;,w)) >■· ·■· > W2(%2(ar,w)-®2(a\w)) > W^few) - Ф^щ)) = 0.

Means and Their Inequalities 127 Remark (ii) (R) seems to occur for the first time in [HLP p. 61] where it is given, in the equal weight case as an exercise and attributed to Rado. Known as Rado's inequality it has been rediscovered many times; see [Bullen 1965, 1970b, 1971b; Bullen L· Marcus; Cakalov 1946, 1963; Dinghas 1943/44; Ling; Popoviciu 1934b; Stubben]. Remark (iii) (P) was proved first in the equal weight case by Popoviciu, [Popviciu 1934b], and is known as Popoviciu's inequality. It was implied in an earlier paper but the author of that paper failed to notice what he had proved, [Simonart]. As in the case of (R) this inequality has been rediscovered many times; see [Bullen 1965,1967,1968; Bullen L· Marcus; Dinghas 1943/44; Kestelman; Klamkin 1968; Mitrinovic 1966; Mitrinovic L· Vasic 1966b, c]. Remark (iv) By adding the inequalities (4) we get another inequality also known as Rado's inequality; η У^ u)kak > Wn<3n(a;w) - Wm<3m{a;w), 1 < m < n. k=m-\-i Remark (v) For an alternative formulation of proof (v) see below 3.4 Remark (и). A point of some logical interest is that the apparently stronger inequalities (R) and (P) are equivalent to (GA) since, for instance, proof (гг) uses (GA) to prove both (P) and (R). Moreover this proof can be used to obtain the following extension of 2.2.3 Lemma 5. Lemma 2 It is sufficient to prove (GA), 2.1 Theorem 1, in the case of equal weights and where in addition α has all terms except one equal. D By 2.2.3 Lemma 5 it is sufficient to consider the case of equal weights, and so since by Remark (i) (R) implies (GA), it is sufficient to prove (R) in the equal weight case. That is, from (3), to prove that ^ + ^^^(α) > αΙ'»<ΰ£_-ϊ),η(α), (5) with equality if and only if an = 0n_i(a). Inequality (5) and the case of equality follows from the case of (GA) that satisfies the conditions of this lemma. D Remark (vi) Since inequality (5) follows from 11.2(7) with χ = (αη/($„_! (α)) and η replaced by η — 1 we get yet another proof of (GA).

128 Chapter II Corollary 3 If a,w_ are η-tuples, η > 2, then 2ln(a;w;)-<Sn(a;;w)>— max laiwi-{-aj'Wj-(wi + Wj)(a^iaJ3) w% Wj |,(6) %п(а;ш) > ( max (maj+Wjaj 1 iV^" /7\ D By Remark (i) the left-hand side of (6) is not less than W η Wn This implies (6) since the η-tuples a, w_ can be simultaneously re-ordered to maximize the right-hand side of (8) and leaving the left-hand side unaltered. Inequality (7) is proved in a similar manner. D Remark (vii) Inequalities (6), (7) give more precise right-hand sides to (1) and (2); they are special cases of I 4.2(8). In the case of equal weights they are particularly symmetric: 2in(a) - 0n(a) > -(л/maxa— ySa) ; —f-^- > max < - + 7 ( h —) > 5η(α) \ι<ϋ'<ηΐ2 44aj ai;) A Rado type inequality involving the arithmetic and harmonic means has been proved; [Klamkin 1968]. Theorem 4 If α and w_ are η-tuples, η > 2, then йп(а;ш) ν -6n-i(a;^) with equality if and only if ап = л/%г-1 (д; ш)<Йп-1 (а; ш) · D = r; τ ч + wn + Wn^n-i " + τ г ^И^_121тг_1(а;щ) 2 Ып-M'w) > ~ / ч— + wn4-2wnWn_1W- -, by (GA) , fin-1 (а; ш) V fin-i(a;WJ ^п + ^п-ц τ г

Means and Their Inequalities 129 which implies (9). The case of equality follows from that for (GA). D Remark (viii) Repeated application of (9) gives a proof of (HA), as well as refinements of (HA) along the lines of Remark (i), inequality (7) and Remark (vii). Remark (ix) Rado-like inequalities between the harmonic and geometric means have been given by Alzer; [Alzer 1989d]. 3.2 Extensions of the Inequalities of Rado and Popoviciu Various authors have given extensions to inequalities (R) and (P). Interesting though these may be they seem to have few applications. For this reason, and since most, if not all, of these extensions can be deduced from a more general result to be given later, full details will not be given in this section; [Bullen 1967, 1968, 1969a,b, 1970b, 1971b; Gavrea L· Gurzau 1986; Mitrinovic L· Vasic 1966a,b,c, 1967a,b, 1968a,b,c]. 3.2.1 Means with Different Weights The most obvious extension is to allow the means in (R) and (P) to have different weights. Unlike the situation for 2.5.3 Theorem 21(a) such results are not immediate consequences of the original inequalities. A particularly fruitful method of discovering these extensions has been used by Mitrinovic & Vasic, [AI p.90]. The proofs are based on proofs (ii) and (Hi) of Theorem 1. Some of these results are collected in the following theorem. Theorem 5 (a) Let λ, μ £ Ш with Χμ > 0, and a,u,v_ be η-tuples, η > 2, with Un > μηη. Then Xv VnZntev) -Un<6£(a;u) > un Vn^^n-i(a;v) (10) - (Αμ)ϋ"/(ϋ"-^")-^-([/η -м«п)б1/хЛп"1)/(С/п"М"")(а;«)· If Un < μηη then (~10) holds. In both cases there is equality if and only if an = (A^)u»/(u»-'ltt»)0Uu"-l/(t/»-'1,1»)(o;u). (b) Let λ,μ > 0, and a, u, v. n-tuples, η > 2, with 2lrl_1 (a; v) + > 0, Vn > μυη Vn — l then (Kn(a;v) + \)Vn > (K(a;u)){v"Un)/u" ~ XVn\v*-wn (11) μμυ" \νη-μνη) (0£_1(a;u))(,",t'n-l)/ttn

130 Chapter II IfVn < μνη then (~11) holds. In both cases there is equality if and only if αη = μ9ίη(α;υ) -f λ. (с) Let α, /3, λ e R with λ > 0, and a, u, y_ be η-tuples, η > 2, with β(α — β4η) > 0. Ιί(α — βυ,η) > 0 then (2in(a;iz) + A)Q " £77 — > (12) If а — βηη < 0 then (~12) holds. In both cases there is equality if and only if (a - βΊΐη)νηαη = /3nn(K-i2tn-ife^.) + АУП). D (а) (г) If λ = μ = 1 (10) reduces to Vn%n{a:,v) -Un<3n(a;u) > K-i&n-ifev) -[/"„.^п-Да; u), un un and proof (Hi) of Theorem 1 suggests putting χ = an and f(x) equal to the left-hand side of this last inequality. The method of Mitrinovic & Vasic that leads to the general case consists of introducing two extra parameters and putting χ = an and g\^(x) = g(x) equal to the left-hand side of (10). Then ^^^-^η^-λμχ^-^/^^Γ^^^η)), and so g'(x) = 0 if χ = ((А/х)^0£_ГЧа;ш)) Under the first set of hypotheses this is a minimum, whilst under the second set it is a maximum. The proof of both cases then proceeds as the proof (Hi) of Theorem 1. (ii) The method of proof (гг) of Theorem 1 can also be used here. However, unlike proof (г) above, this method gives no help in discovering the correct form of the inequality to be proved. Simple manipulations show that (10) is equivalent to -γΓαη + {(Χμ) ηΚ-ι (йЛШ >ar?n \μ<δ*_ΐ η(α;η), (13) which is an immediate consequence of (GA).

Means and Their Inequalities 131 (b) The proof of (11), of which (P) is a special case, follows using either of the methods used in (a). However it is of some interest to note that (b) is the same as (a). Putting λ = —^r^^nfe^) — 21η(α; ν), μ = n n/3, (11) reduces to (10), with λ, μ replaced by α, β respectively. (с) As in (a) or (b) put X Οιγι and consider f(x) = — ■——. Then / has a unique extremal at χ = β , which is a minimum if vn α- βηη the first hypothesis holds, and a maximum if the second holds. D Remark (i) It might be noted that (13) shows that (10) is independent of v. Example (i) We will see later, IV 3, that (R) and (P) are special cases of a more general result. Here we note that the single inequality (10) also includes them both by choosing special values of λ and μ. Take λ = μ = 1 and и = У_ = Ш then (10) reduces to (R); taking λ = 2in(a;i^)/^nfeuz), and μ = l,u = y_ — Ш> (Ю) reduces to (P). Example (ii) If a = Vn, β = vn/un then (12) reduces to {^n(^v) + x)Vri (^η-ι(α;^) + λ^-) which is a multiplicative analogue of (10). Remark (ii) Inequality (10) could be further extended by considering 9λ,μ^θ{χ) = 9{x) = Vn%n(a',w) - A0£(a;;u) - И£>£(а; v) where χ = an and 9unVn = μνηυη. Various choices of the parameters give results obtained independently by several authors. Example (Hi) Taking λ = μ = 1 inequality (10) reduces to a particularly simple extension of (R) due to Mitrinovic, [Mitrinovic 1968c}: —2ln(a; v) -6n(a;u) > -^Stn-^a; v) !iz:i0n_1(a;n). vn un vn un Example (iv) Taking λ = unVn/vnUn, Χμ = 1 in (10) gives the following inequality that is another generalization of (R): Vn(%n(a;v) - K"Un/UnVn(&^) > Vn-i(bn-i(a;v) - К"-Г lM"-'(ai3i)).

132 Chapter II Remark (in) For further extensions see [Wang· С L 1979a,1984c]. 3.2.2 Index Set Extensions Let a, w be sequences and I an index set, I € X, see Notations 4, then extending the notations of Notations 6(xi) and I 4.2 Theorem 15, we write %i(a,;w) = — 2^ВД; ® ι (aim) = \[[wiai) Using this notation (GA), (R) and (P) can be expressed in the following form. Theorem 6 Let a, w_ be sequences and define the following functions on the index sets, I el, p(I) = WI(M^m)-®i(a;w)), тг(7) = (^Щ) ^ ; then both ρ and π are non-negative increasing functions. The results of this section are concerned with obtaining more properties of the functions ρ and тг; [Bullen 1965,1967; Everitt 1961; Mitrinovic к Vasic 1966b]. Theorem 7 The set function ρ is super-additive, and the set function π is log- super additive. Remark (i) The first part of this theorem is a special case of I 4.2 Theorem 15(a), and both parts follow from more precise results which we now give. If α is an (n + ra)-tuple, n, m e Ν*, α = (al5..., αη+πι), we will write n-\-m n-\-m α= (an+1,...an+m); 21т(а;ш) = -^=r- ^ wiai\ Wm = ^ w{; etc. (14) Theorem 8 If a and w_ are (n + m)-tuples then W (a) Wn+m(2tn+m(a;^) - 0n+m(a;;w)) > Wn{%n{ar,w) - <3n(am)) + Wm(^im(a;w) - <Зш{а;Ш)) ®п+т(а;шу \®п(а',Ш/ \0w(a;w There is equality in (a) if and only if <Sn(a;w) = ^mfe w); and there is equality in (b) if and only if 2ln (a; w) = %ш (a; w) · D 3.1 Theorem 1 is a special case of this result, m = 1, and proof (ii) of that theorem can be used to prove Theorem 8. Π

Means and Their Inequalities 133 Remark (ii) This result has been extended to allow for different weights in the different means; [Bullen 1967; Mitrinovic Sz Vasic 1966b]. Remark (iii) The inequality in 2.5.3 Theorem 21 (b) also has index set extensions; see [Dragomir L· Goh 1997a]. 3.3 A Limit Theorem of Everitt Given a sequence α then the equal weight case of (R) says that the sequence n(2tn(a) — Φη(α)), η Ξ Ν, is increasing. It follows that limn^oo n(2tn(a) — Φη(α)) exists, in R, and it is natural to ask under what conditions this limit is finite. Everitt gave a complete answer to this question, and in this section we present his work; [Everitt 1967,1969]. Theorem 9 If α is a sequence then limn_>oo n{pin(a) — Φη(α)) is unite if and only if: either (a) Σ™=1 α% converges, or (b) for some а > 0, Y^l^di —a)2 converges. D The proof is given by considering the four possible types of behaviour of a: (a) limn^oo an = 0; (β) limn_>oo an = a, a > 0; (7) limsup^^ an = 00; (δ) 0 < liminfn^oo an < limsup^^ an < 00. (a) Using 3.2.2 Theorem 6, and the notation defined there, if 1 < к < η, η Σ <Ь > η(<Άη(α) - βη(ο)) =ρ({1,..., η - 1, η}) > ρ({1,..., к, η}) ί=1 i=i i=i Letting η —> oo, and then letting к —► сю we get that limn^oo n(2ln(a) — 6n(a)) = Σϊι аь which completes the proof in this case. (β) First remark that this hypothesis implies: 1 n lim 2ln(a) — a, and lim — / (α^ — α)2 = 0. n—voo n—юо П τ—-/ г=1 Now define r^· = a^ — 2lj(a); obviously Σ\=1τ^ = 0, and we investigate the properties of the two quantities ^ τ2·, ^ r^·. i=i i=i Now: i=i i=i i=i к к ^ Σ7* = Σ (^ "«) + («- Slife))2» * < k < 3-

134 Chapter II From this, and the preliminary remark, we make two deductions. First, letting j —► oo, and then к —► сю, we get that lim £>S = X>i-a)2; (15) also, 1 j lim -Vr?, = 0. (16) Now, given б > 0 choose n0 such that if η > n0 then |an — a\ < e. If j > n0 then no j i г—ι г=гго + 1 г=1 Hence Σ^· = °(Σ^).^~- (18) i=i i~i If we assume that Х^Да* — «)2 = сю, then using (15), (17) implies that Σ4 = °(Σ0>^~· (19) We now proceed to consider the proof of the theorem in this case. βηω =*„ы ft О + вд)1/η=*»ω«ρ Й Σы* (ι + ^)) .зде^Аф^-^ + осы»))) =<М»)еч> ( - Ι —(£,& + 0(£ы»))). nV—/ г — χ ^ = 1 If then Σ~ι(α» - α)2 < oo then from (15) and (18) м,)=адехр(4^|>4 or η (81η(α)-β„(α)) = ^^Στ^ + 0(1)>

Means and Their Inequalities 135 so limn^oo n(2tn(a) — Фп(й)) < oo. If, on the other hand, Υ^Ι^α,ί — a)2 = oo, then from (15) and (19), -l + o(l) ^^w^tw?^)· η (α) or »(«ηω-β»ω) = ^Σ^ and so by (15) limn_>oo w(2ln(a) — Φη(α)) = °°- (7) Assume, without loss in generality, that limn^oo сьп = oo, then by 3.2.2 Theorem 6 p({l,..., n}) > p({l, n}) =a1 + an- ^/axan, and so limn^oo n{pin(a) — Φη(α)) = сю. (δ) Let /? = liminfn^oo an, β = lim suprwoo an and e = (β — β)/3; further suppose that 0 < an < λ, η e Ν*. Now choose two sequences of integers p, g such that if г Ε Ν* then Pi <qi < Ρ%+ι; αρί < β_ + e; aqi > β- е. Then aPi -agi>e,iGN*, and so р({Ръ0*}) =aPi +aqi - l^pi^i > jy (20) So by 3.2.2 Theorems 6 and 7, and (20) p({l, ■ · ·, 0*}) >p({Pi, 0i, ■ · · ,P», %}) >p({Pb 01, · ■ ■ ,P*-l, 0*-l}) + p({p», 0*}) * · 2 > Σρ({ρ*>0λ}) - τ; fc=l and so limn^oo w(2ln(a) ~~ 0η(α)) = oo. D Remark (i) Everitt's original proof of case (7), [Everitt 1967], contained an error that was pointed out by Diananda in his review of this paper, [Diananda 1969]. Diananda later supplied a correct proof; [Everitt 1969]. Remark (ii) It might be noted that this theorem gives some information on an upper bound for 2ln(a) — 0n(a); this topic is taken up in more detail later; see 4 below.

136 Chapter II A completely different limit is that given by the following result of Alzer, [Alzer 1994a]. Let fc = {l,2,...,fc}, k = 1,2,..., then lim η^Γ7ΐτ ~ (n ~ !); ~ n-oo \ (5n(n) 'gn-xfn-l)/ 2' The author has given other results for the means of this special sequence; [DI p.18], [Alzer 1994b,c,d, 1995b]. 3.4 Nanjundiah's Inequalities In this section we establish a very interesting extension of (GA), inequality (31) below. Given two sequences a, w_ let us write Я(а;ш) = и = {Slifem), »2(а;щ), }; 0(а;ш) = g. = {^1(а;ш), ^2fe ш), }, for the sequences of arithmetic and geometric means. Regarding α ι-> 21, α ι-> 6. as two maps of sequences into sequences Nanjundiah's ingenious idea was to define inverse mappings as follows:17 ^Ча;ш) = -«η - ^α„-., e-^fijai) = t , , η e Ν*. η—ι These will be called, respectively, the sequences of inverse arithmetic and geometric means of α with weight w, and we will use the above notation to write 3t_1few;) = a-1 = {Щ1(&ш),Щ1(а;ш) ...}, As usual in case of equal weights reference to the weights will be omitted in these inverse means. These inverse means have the elementary properties (Co), (Ho) and (Re) of the original means; see 1.1 Theorem 2. In addition the obvious analogue of 1.2(8) holds, the inverse arithmetic mean is (Ad) and the inverse geometric mean has the property 1.1(9). Lemma 10 (a) With the above notations МЗГ1; ™) = а^ШЫ = 0п(£-1;ш) = ®пЧ&ш) = ап, п е Ν*. (Ь) If η > 1, ®-Ча]Ш)>Ъ-\аЖ), (21) 17 Remember that Wo=0] see Notations 6(xi) See earlier 2.4.3 proof (xx), 3 1 Theorem 1 proof (v).

Means and Their Inequalities 137 with equality only if αη_ι = an. (c) If η > 1 then ®п\йж) + ®п\кш)>®п\* + ки1), (22) with equality only ifanbn-i = an_ibn. D (a) Elementary. (b) If we put a = Wn/wn then a > 1, and if further we put a = an, b = αη_ι, (21) becomes (B) in the form I 2.1(2). Alternatively we can note that 21"1 and 0"1 are arithmetic and geometric means with general weights and use 3.7 Theorem 23. (c) By the elementary properties noted above and (b) we have β„ (a + b;w) > a;1 (a/(a + b); w) + Я"1 (b/(a + b); w) = Ш-\а/(а + Ь) + b/(a + b);w) = 1. The case of equality follows from that of (b) Remark (i) Clearly (a) justifies the name inverse mean given above. D Remark (ii) Inequality (21) is an analogue of (GA) for these means and can be used to prove both (R) and (P). In fact proof (v) of 3.1 Theorem 1 is just two applications of (21) with a replaced by И, and then by <3; wn wn Remark (Hi) Inequality (22) is an analogue of III 3.1.3(10), and will be used to give a proof of that result, III 3.1.1 Theorem 8. It is natural to consider what happens when the means in Lemma 10 (a) are interchanged. Lemma 11 Ifn> 1 and ifWnan, η £ Ν*, is strictly increasing, and Wn/wn, η Ε Ν* is increasing then вй1(3~1;ш)>Я£1(£-1;«>), (23)

138 Chapter II with equality only if an_2 = an-i — αη· D On writing out (23) we have to prove that Wn _Wn-1 Wn Wn Wn/wn Wn-1/wri (24) W I aWri/Wn \ W iaWn~l/w "n—1 / \^Π—2 Rewriting (24) with a simpler notation what we have to show is that (т"(Г"1)СГ >^-(r-D^r, (25) («с-(«-1Г ~ cr b"~ and the conditions of the lemma imply the restrictions ra > (r — l)c, qc > (g — l)b, r > 1, q > 1, and r > q. The case q = 1, which occurs if η = 2, is an immediate consequence of (~J), I 4.2 Remark (i), and the convexity of xr, χ > 0, r > 1. So we now assume that q > 1. Let us define β by re — (r — 1)/? = qc— (q — 1)6, when the left hand side of (25) becomes (ra — (r — l)c) (rc-tr-l)/?)"-1' which by (22) is greater than or equal to (raY ((r - l)c)r (26) (27) (re)-1 ((r _ i)/3)'-i * Now by (GA) 0 = ^=4c+ ^—U > (с'-вьв-1)1/(г-1), or equivalent^ r — 1 r — 1 7 cr cq Collecting (27), (26) and (25) we have proved (24). For equality in the use of (GA) we need that b = c, when β = с. For equality in the application of (22) we then need a — c. This completes the proof of the lemma. D Remark (iv) The reasons for the restrictions on a and w_ are clear from the proof: (i) the left hand side of (23), or equivalently (24), needs Wnan to be strictly increasing; (ii) for β to be an arithmetic mean the weights must be positive; and

Means and Their Inequalities 139 the condition Wn/wn strictly increasing ensures that r — q > 0. While if Wn/wn is increasing then the case r = q means that β — h and the discussion is simpler. Remark (v) When Lemma 11 is applied to different sequences a we must check that the first condition holds for those sequences. In particular, in the case of equal weights the first condition reduces to nan being strictly increasing and the second is satisfied. Theorem 12 If η > 1, and ifWn/wn, η e N*; is increasing then ,Яп(Й;ш)>/ ~ \*n-i(&;w)) ' (29) with equality only if an = Φη_ι(α;υ;) = 2ln-i(i£;u;). D Since Wn$ln(a; w) is strictly increasing we can apply Lemma 11 to 21 to get, using Lemma 10(a), In other words W^(2tn(0_1;w) - &~\^w)) is an increasing sequence. This implies that sinifi"1;^^^-1®^). (30) In (30) replace a by (6 to get which completes the proof since the cases of equality follow from those of Lemma 11. D Corollary 13 If η > 1 then 6n(5l;^)>2ln(0;^), (31) with equality only if α is constant. D Rearrange , if necessary so that Wn/wn, η £ Ν* is increasing. This makes /<3n(2l;w)\Wri no difference to ( ™ '—г , which by (29) exceeds 1. D Remark (vi) Theorem 12 is a Popoviciu type extension of (31) and we can also prove a Rado type extension by a similar argument. All we have to do is to start the proof of Theorem 12 by applying Lemma 11 to the sequence j6_. For this however we need an extra lemma since it is not immediate that Wn<3n(a;w) is strictly increasing.

140 Chapter II Lemma 14 IfWnan is strictly increasing then so is WnQ5n(a;iii) provided Wn/wn is increasing. D Note that Wn<3n(a;w) = ®n(a;w), where ( WWi 1 n *"' if 1< t < n, ΙαΓ if г = 1. Then a^ = WidiPi, where Wi-i/wi ll if г = 1. By hypothesis Wn/wn is increasing and so βη is increasing by I 1.2.2(11). Hence an is strictly increasing and so therefore is WnGn(a; г/;). D Theorem 15 If η > 1, Wnan, η £ W, strictly increasing, and Wn/wn, η e N*, is increasing then Wn (0п(а;щ) - Яп(Й;ш)) > Wn-ι (вп-1@;ш) - Яп-1(^;ш)), witJh equality only if αη = 21η_ι(α; υ;) = Фп-гО&ш)· Nanjundiah used the equal weight case of (31) to prove the important classical inequality, Carleman's inequality. This inequality has been the object of much research; for further information see [AI p. 131; DI рр.Ц-^5; EM2 p.25; HLP pp.249-250], [Pecaric L· Stolarsky]. Other proofs of this inequality are given later, see 3.6 Remark (i) and IV 4.2 Remark (ii). Corollary 16 [Carleman's Inequality] If α is a non-null sequence then η η Яп (β(α)) < е21η(α), or £) «η(α) < е£) «г, η € Ν*; if further Σ°11 а < oo then oo oo tJhe constant is best possible.

Means and Their Inequalities 141 η D The equal weight case of (31) can be written as /^Φη(α) < ®n(l)> i=i Vn! where 0n(s) is the geometric mean of the first η terms of the sequence s = {αϊ, α\-\- 0,2, αϊ + ci2 + аз, · · ·}· However η/ \[гй < е, I 2.2, and so η η ^0η(α) <e<3n(s) <e^ai- г=1 г=1 The rest follows as in [ЯЬР]. D Remark (vii) All the results of this section are stated in [Nanjundiah 1952], and proved in the author's unpublished thesis that was communicated privately to the author who then published Nanjundiah's proofs; [Bullen 1996b]. In the meantime, based on numerical results, the equal weights case of (31) was conjectured. A proof of this conjecture was given by Kedlaya, [Kedlaya 1994]. After the appearance of this paper several other papers appeared, all apparently without knowing of Nanjundiah's work. In these papers alternative proofs were given for Kedlaya's result and also of the general inequality (31); further various extensions were made; [Holland; Kedlaya 1999; Matsuda; Mond L· Pecaric 1996a,b; Pecaric 1995a], see III 3.2.6, VI 5. Remark (viii) These results can be regarded as mixed mean results; this topic is taken up in III 5.3. 3.5 KOBER-DIANANDA INEQUALITIES Throughout this section α is an n-tuple that is non-constant and non-negative, υ; is a positive η-tuple with Wn = 1. Further we will use the following notations: w = minw; W = maxu>; Δη = 21η(α; w) - 0η(α; w); Dn = Sin (α) - #η(α); Ση = Σ WiW5{y/oi- να")2; Sn = ^ (^/ο~-^α~)2. i<i<j<n i<i<J<n It is worthwhile noting some other forms for Ση and Sn. 1 η η η i,J=i 1 η η η i^^^-^)2-^-1)!^2 Σ ν^4· (33) 2 i,j=i i=i i<i<j<n

w η — 1 1 Δη <—- <W: <Ι. 142 Chapter II Theorem 17 (V) With the above assumptions and notations (34) 1-и;^Еп-«/ (35) Cb^) There is equality on the left-hand sides of (34) and (35) only if either η = 2, or η > 2 and the αι with minimum weight is not zero, and the rest of the a^ are all zero. (c) There is equality on the right-hand side of (35) only if either η = 2, or η > 2 and the a; with minimum weight is zero, and the rest of the α{ are all equal and non-zero. The upper bound of (34) is in general not attained. D (г) The left-hand side of (34)· Using (33) we see that, Π П су П Δη TSn=y^Wiai-wy~]ai-\ У] уЯ~а~ - <6n(a;w) η—Ι έ—' ζ—' η —1 ^—' ν J i=i i=i i<i<j<n η су η = Y](wi - w)a,i + Υ] y/^Oj-®n{a',m), Ύϊ — 1 i—i i<i<j<n where we have assumed without loss in generality that w\ = w. Now the two sums in the last line contain terms involving the a^, 2 < г < η, and the yfdidj, 1 < г < j < n, with weights that sum to 1; that is *%2™=2(щ — w) -f (!*Ll2))(jHL) = i. Hence applying (GA) we find that these sums exceed w ®п(а]Ш.), so that Δη -Sn > 0, which is equivalent to the left-hand side η — 1 of (34). Further by the case of equality in (GA) this inequality is strict, the a not being constant, unless η = 2, or if η > 2, a\ =£ 0 and all the other a^, 2 < i < n, are zero. (гг) The left-hand side of (35). The proof in this case is similar to that of (г), but using (32). 1 1 η η Δη - Ση =- ( У^ WiWj,/a~aJ -w Υ^ιυ^αΔ - <бп{о±',ш) 1 — w 1 — w V ,^-—' v ^-^ / *j=i г=1 1 η η =——( ]P WjUij^ajaj + y^jwi(wi - w)aj) - 0п(а;ш), where again, we have assumed without loss in generality that w = w\.

Means and Their Inequalities 143 As in (г) the sums form an arithmetic mean, and an application of (GA) shows that Δη — Ση > 0, which is equivalent to the left-hand side of (35). 1 — w The cases of equality follow as in (г) . (Hi) The right-hand side of (35). Using (32) this is equivalent to proving 1 n Φη = Φη(α) = - Σ wiWj(y/a~i- у/а]) - w(%l(a;w) - 0η(α; w)) > 0. (36) The proof of (36) is by induction on η and since the case η = 1 is obvious assume the result for all k, 1 < к < п. Assume, without loss in generality that an = mina, put w = min{u>i,..., u>n_i}, and 7 = Φη-ι(α;2£')> where Ш = (ϊ ,··■>! )· Then Φη(7,7,...,7,αη) = (wn -w)(l - wn)j + wn(l - w - wn)an + WJ1~Wna™rι -2wn(l -wn)y/ja^ and by (GA), Φη(7,7,...,7,αη)>0. (37) Now if a/ = (ai,... , αη-ι,21η-ι(α; w'))> and 1 n —1 Φη-ι(α') = 2 Σ ^jl^"^)2"!^ (%η-ι(α;υ/)~7), then by the induction hypothesis Φη_ι(α') >0. (38) Further <$>η(α)-Φη(Ί, 7, ■ · ·, 7, «η) - (1 - ^η)2Φη-ι(αΟ η—ι =(гу - ги + wn) ( ]Ρ ги»а» - (1 - wn)jj η —ι - 2υ;ηλ/α^ ^ ^ν7^ - ί1 - ™п)л/7) · ί=1 Now by (GA) both of the terms in the brackets are non-negative. Hence using (37) and (38) n—1 n—1 Φη(α) >юп(^м»а» ~ (X ~^n)7j - 2™n>/7( X] wiy/<H ~ i1 ~^n)y/lj, i=i i=i n — 1 =tw„ Σ MV^i - л/7)2 > О. (39)

144 Chapter II This completes the proof. Further there is equality in (39) if and only if a^ = · · · = an_x =7. Inequalities (37) and (39) are equalities together if and only if w = wn, an = 0 and ai = · · · = CLn-i > 0. (iv) The right-hand side of (34)· We can write Sn = nDn+Fn where Fn = η2Φη, and Φn is as above but with equal weights. Hence by (39) £s ;■ (40> Further, assuming without loss in generality that wn = W, D--^ = ^^W-w^ + ^(a;m)-en(a)>0, by (GA). So ^ < nW. (41) Combining (40) and (41) completes the proof of this case. The final remark in (c) is shown by the following example. If we take η = 3 and wi = u>2 = 1/4, гУз = 1/2, then the upper bound of As/Ss occurs if either αϊ = 0, α2 = (\/3 — 1)2аз, or а2 = 0, αχ = (\/3 — 1)2аз, when If we put r = 2(1 Л—-=) what has to be shown is that v3 (r - 2)a3 + (4^aia2- Kv^+ V^))^+(r - !)(ai + a2) - 7"л/а1а2 > 0. (42) This holds, since the coefficient of аз is not positive, if and only if (4^oT^ - г(л/аТ + χ/αϊ))2 - 4(r - 2) ((r - 1)(αι + a2) - гл/аТа!) < 0. Since 3r2 — Ylr 4-8 = 0 this is equivalent to ν^αια2(>/αι + л/а^) > 2λ/αια2, (43) which is obviously true. For simultaneous equality in (42) and (43) we must have that 2(r — 2)л/а3" = — ^α\α<ι 4- r{y/a{ 4- i/ai") and αϊ = 0, or a2 = 0, or a\ = a2, and since not all of αϊ, α2, аз are equal this completes the proof. D

Means and Their Inequalities 145 Remark (i) The result (34) is due to Kober; Diananda proved (35) and gave the above simplified proof of Kober's result; [Diananda 1963a,b; Kober]. A multiplicative analogue of Theorem 16 has been given, where the differences An,Dn are replaced by ratios; [Bullen 1967). Remark (ii) Both Kober and Diananda gave consideration to the behaviour of the ratios An/Sn, and Δη/Ση as α tends to a constant η-tuple; in particular if the limit is not the zero η-tuple then Δη/Ση tends to 2. Remark (iii) The following inequality can be found in [Beck 1969a}] see also [Motzkin 1965b). (mina)3 ^ Λ h2 ®п(а;ш) sr^ , x2 Remark (iv) Some of these inequalities are related to the mixed mean inequalities of III 5.3; see [Mitrinovic L· Pecaric 1988b]. In particular the inequality Fn > 0, see the equal weight case of (39), is equivalent to (п-2)9Яп(1,0;1;а)+Ш1п(1,0;п;а) > ЯЯП(1,0; 2; а), using the notation of III 5.3 Remark (v) The inequality (39), Φη > 0, or equivalently 1 n 2tn(a; WL) ~ ®п(а;ш) < γ- ^ WiWj(y/al - y/aj) and the above inequality of Beck, are examples of converse inequalities, a topic discussed in more detail later, see 4 below. Remark (vi) Alzer has refined the right-hand side of (34); see [Alzer 1992a]. Remark (vii) Aczel, in his review of the above paper by Alzer, [ZbL, 0776. 26013], pointed out that the Kober-Diananda inequalities are helpful in situations where the geometric mean is needed but the arithmetic mean is easier to calculate; [Aczel 1991]. 3.6 Redheffer's Recurrent Inequalities A very interesting general method for obtaining inequalities is in [Redheffer 1967, 1981]. The method leads to certain refinements of (GA) but has much wider applications.

146 Chapter II Definition 18 Let /&, 1 < к < η, be intervals, and Д, g^ : Π£=ι ^ l~* ^> /^ ^ ^> 1 < /c < n, then η η Ι>Λ<Σ^ (44) i=i i=i is called a recursible or recurrent inequality if there exist functions Ffc : R и R, 1 < к < η, such that Ή:(μ)Λ-ι(α*,·· -,afc-i) = sup {μΛ(α*, ..., afc) - pfc(a*, ..., afc)}, where /o is denned to be 1. A good motivation for this definition is given in the first reference. Theorem 19 The recurrent inequality (44) holds for all α G ΠΓ=ι ^ if and only if there exist 4^1, l<fc<n + l, £η+ι = 0, δι < 0, such that Mfc = -^fc-1№) - tffc+i, 1 < /с < ?г; where for each k, 1 < к <n, Fj^1(S) being any one of the solutions of -Ffc(/i) = δ. D The proof is by induction on n; since the case η = 1 is obvious let us assume the result for all k, 1 < к < п. The inequality will hold for к = η if and only if it holds for the unfavourable choices of an, assuming αϊ,..., αη_ι fixed. Since the inequality is recurrent we get, using the induction hypothesis, the relations of the theorem for /с, 1 < к < η — 2 and also Fn^n) + /in- = F'l^Sn-!); and defining δη = Fn(/in), completes the proof. D As an application we give the following result. Corollary 20 If β^ 1 < к < η 4- 1, are non-negative, βι < 1,/?η+ι = 0, and λ^, 1 < к < η, positive then η η Σ>((λ*/?*)1Α -/?££) 0*(α) < £λ*α*. (45) fc=i k=i η η D The inequality Υ^μ&0&(α) < У^^а^ is recurrent. M^fc(a) - Afcafc = (/it - Afctfc)0fc_i(a), To see this consider

Means and Their Inequalities 147 where tk = α&/($&_ι(α), к > 1. We then obtain that, к > 1, I 0, if μ < 0; and ί\(μ) = 0, μ < λχ, ίι(μ) = oo, μ > λχ. Since ί&(μ) > 0 implies that 6k > 0, put 5k = {к — l)/?fc ~ to get that where the /3fc,l</c<n — 1, are as stated. D Corollary 21 If 2in(6) = 2ln(0i(a), 02(a),..., 6n(a)) then ^(ejexp^^^^eOnia), (46) and -2 U„(a) + 2ln(®)/~ β„(ο)' l j with equality if and only if α is constant. D Apply Corollary 20 with pk = ei(fc_1), t > 0, 1 < /c < n. By the mean-value theorem of differentiation, see I 2.1 Footnote 1, ΚβΙ'" - β\'+\) >-te\ so, taking λ^ = 1, 1 < к < η, (45) leads to 0η(α) < 21η(α)β-* + *An (0). (48) The best choice of t is (βη(α/Λη(^)) - 1, which gives (46). Now assume, without loss in generality that, α is increasing, and is not constant; then 0 < 0i(a) < 02(a) < ■·· < 0n(a), and An(0) < 0n(a). So (47) follows from (46) using the inequality in I 2.2 Remark (ii) and (20). D Remark (i) Both of the inequalities (46) and (47) sharpen (GA). Since the exponential term on the left-hand side of (46) is bigger than 1, (46) implies Carleman's inequality, 3.4 Corollary 16; see also [Alzer 1993c]. Remark (ii) A direct proof of (48) is given in [Bullen 1969c] using methods employed earlier in this chapter. Unfortunately the method requires that 0 < t < 2 and so does not give (46).

148 Chapter II 3.7 The Geometric Mean-Arithmetic Mean Inequality with General Weights Clearly if all the weights are negative nothing new occurs since we only use the positive ratios Wi/Wn, 1 < г < η. Further if we allow non-negative, or non-positive, weights, with of course not all being zero, the effect is to reduce the value of n, and to change the condition of equality from "a is constant" to "a is essentially constant", see I 4.3. We will further talk of the arithmetic mean being essentially internal meaning that in 1.1 (2) the minimum and maximum is taken over the essential elements of a, with analogous usages for other properties. In the case of (R), 3.1 Theorem 1, the extra generality means that either wn = 0 when the inequality is trivial, or wn =fi 0 when as remarked above the result includes all the cases 2 < к < η and the case of equality is unaltered. However if both positive and negative weights occur (GA) can still hold; [Ayoub]. This is a consequence of 2.4.2 proof (xvi) which shows (GA) is just a special case of (J). Hence applying the Jensen-StefFensen theorem, I 4.3 Theorem 20, instead of (J) we get the following result. Theorem 22 If η > 3, a and increasing η-tuple, and w a real η-tuple satisfying Wo Wn^0, and 0 < -1 < 1, 1 < i < η (49) then mina < $ln(a;w) < max a, mina < 0n(a; w) < max a. (50) and (GA) holds with equality if and only if α is essentially constant. D The proof of the left-hand inequalities in (50) is given in the preliminary remarks in the proof of I 4.3 Theorem 20; the right-hand inequalities can then be deduced in a similar manner, or by using 1.2 (8). The proof of (GA) follows 2.4.2 proof (xvi), but now using the Jensen-Steffensen theorem. D Remark (i) All the methods used to prove I 4.3 Theorem 20 can be used to obtain this result; the proofs are now simpler as this is a very special case of that theorem. When η = 2 the situation is different and very elementary but is worth stating for reference; see also 2.2.2 Lemma 4(b). Theorem 23 If η = 2, wiw2 < 0, W2 = 1, α a 2-tuple then (~GA) holds. More precisely, if a\ < a2 and w2 < 0 then Ща;ш) < ®(a;w) < mina (51)

Means and Their Inequalities 149 while if a\ < 0,2 and w\ < 0 then ®(a;w) > Ща',Ш) > max α. (52) D This follows as a particular case of I 4.2 Remark(i) but we will give a simple direct proof. Assume, without loss in generality, that Q<ai=x<a2 = y, and wi = 1 — t, u>2 = t, t G R. Define the three functions: A(t) = Ща;ш) = (1 - t)x + ty, G(*) = в(а;ш) = ж1-*?/*, D(t) = A{t) - G(t). Now if t < 0 then G?(£) = х{у/х)ь < ж, which give the right-hand side of (51), while if the 1 — t < 0, equivalently t > 1 then A(i) = (1 — t)(x — 2/) 4- 2/ > 2/, giving the right-hand side of (52). Now we easily see that D(0) = D(l) = 0 and D"(t) < 0. So D is strictly concave and hence is negative outside the interval [0,1]. This completes the proof. D Remark (ii) A particular case of arithmetic and geometric means with general weights that do not satisfy (49) is discussed below in 5.8. Example (i) A simple use of Theorem 23 is to prove inequality 2.4.6 (34); let χ be an increasing η-tuple then η η χι Y[{kxk -к- lxfc-ι) <^i[] XkXk-ki~1} = xn- k=2 k=2 3.8 Other Refinements of the Geometric Me an-Arithmetic Mean Inequality There are many other refinements of (GA) of which we will mention a few. Theorem 24 [Siegel; Hunter J] If η > 2 and α a non-constant η-tuple then: (a) ®п(Ь]Ш)®п(д;Ш) < ^n(а;Ш)> (53) where b is the η-tuple defined by b^ = 1 -f (n — k)/(t + k — 1), 1 < к < n, and t is the unique positive root of йП(^Г" п (*-·,)·-(п-П <м> k=i i<i<j<n i=i ®n(a; w) < ^(l + tn-l^l-i)"-1 2l„(а; ш), (55)

150 Chapter II where 0 < t < 1, and t is a root of the equation £ (ai-aj)2=t(n-l)(J2*i)2- i<i<j<n i=i Remark (i) Since for all k, 1 < к < η — 1, Ь& > 1 we have that 0n(b) > 1, (53) is a refinement of (GA). Remark (ii) Eliminating ΠΓ=ι ai fr°m (53) and (54 ) leads to the inequality я*»-"(а)> π κ-^2π^ΓίΡ i<i<j<n k=i ^ ' which in the case t — 0 is a refinement of a result of Schur; see [Schur 1918]. Remark (iii) The results in [Dinghas 1953) follow from (55). Theorem 25 [Fink L· Jodheit 1976] If α is an increasing η-tuple then 0η(α)<21η(α;^), (56) Wn k<i where Wi = —- Π (^ ~ (^ ~~ (— )* )j' 1 — ^— n' w^n equality if and only if α is constant. Remark (iv) The essence of this result is that the geometric mean with equal weights is less than the arithmetic mean with certain smaller weights. Remark (v) A general class of weights w for which (56) holds has been studied in [Fink 1981]. Theorem 26 [Sierpinski's Inequality] If α is an η-tuple then S)n(o) ^ /ЗДГ > (ЯпЩ"-1 j (57) Sin (a) " \%n{a)J ~ V^n(a) with equality if η = 1, 2, or η > 3 and α constant. D The case η = 1 is trivial and for η = 2 see 2.2.1 Remark (i); so assume that η > 2. A simple application of 1.2(7) shows that the left-hand inequality in (57) implies the right-hand inequality, and so it is sufficient to prove that "ОД-" (58)

Means and Their Inequalities 151 The proof is by induction, and let χ = αη, and then put the left-hand side of (58) equal to g(ж), and / = log og Simple calculations show that / has a unique minimum at the point χ — x' where χΙ = (п-1)Ип(й)-Яп(о). since n > 2 we have by (HA) that x' > 0. So for all χ =£ x',g(x) > g(xf). Simple calculations, and a further application of (HA) show that q(x') > n~1 ~ ,η^ ~ and so the result follows from the induction ©r;(«) hypothesis. D Remark (vi) The result is in [Sierpinski]; this proof is from [Mitrinovic L· Vasic 1976], and clearly gives the following Rado type extension of (58): К^ШЛяд ^ aCi(a)fin-i(q) ®n(«) - K-H&) with equality if and only if α is constant. Remark (vii) A simple proof giving the cases of equality can be found in [Alzer 1989a); the same author gives a generalization of (57), [Alzer 1989b, 1991b). An extension to weighted means, when the weights are decreasing, can be found in [Pecaric L· Wang; Wang С L 1979e). The following result generalizes these and inequality (57); [Alzer, Ando L· Nakamura). Theorem 27 If oLis a non-constant η-tuple, η > 2, and w_ is a decreasing n-tuple, strictly if η = 2, define fix) = (en(a1-x;w))Wn(^(a';w))W^4^n(axiw))Wn, ieK. If a is increasing then f is strictly increasing on) — oo, 0], while if α is decreasing f is strictly increasing on [0, oo[. In particular we get the following generalization of (58). Corollary 28 Ifa,w_ are decreasing n-tuples, n>3, then effete) with equality if and only if α is constant. D Take χ = 0,1 in Theorem 27. D Remark (viii) It is clear that it is sufficient to require that the η-tuples a, w be similarly ordered; see I 3.3 Definition 15.

152 Chapter II Remark (ix) The case, χ = r, r > 0, of Theorem 27 is given in III 6.4 Theorem 9. Day kin & Eliezer have given a convex function whose value increases from one side of (GA) to the other, thus providing many refinements of this inequality; [Daykin Sz Eliezer 1967]. As these results follow from I 4.2 Theorem 18, we only give a later result of Chong Κ Μ. Theorem 29 [Chong Κ Μ 1977] If α is a non-constant η-tuple and A{x) = %п(ахК~Х&ш);ш), 0 < χ < 1, then A is strictly increasing. Remark (x) This result generalizes (GA) since A(0) = £5n(a;w) and A(l) = 2ln(a;w). This result has been extended in [Pecaric 1983-1984) where it is shown that Φ η (x 2i-nfe Ш) 4- (1 — x)Q/, w) 2ln [x 2ln (a; w) 4- (1 — x)a; w) is an increasing function: and Chong Κ Μ has given another simple function with a similar property: А/ ч ч \Wi/Wn [[[х%г(а;ш) + (^ ~ x)ai) i=i Remark (xi) For further results of this type see III 2.5.4. Theorem 30 [Wang С L 1980d] If α > Ο,β > 0, η > 0, α and w_ n-tuples with Ql < y/l/β, and if f(x) = (βχ 4-7Ж_1)а then )) < /(вп(а;ш)) < 21п(/(а);ш). (59) D Since / is decreasing on [0,1/7/'β], see I 2.2(f), the first inequality in (59) follows from (GA). Further д = f о ехр is convex so the other inequality follows from (J) and 1.2(7). D Remark (xii) The original proof of Wang used the ideas from linear programming as well as Lagrange multipliers, see 2.4 Footnote 10; the above simple proof is due to Pecaric. Remark (xiii) Inequality (59) generalizes a result of Mitrinovic & Djokovic, see [AI p. 282), and reduces to (GA) ifa = 7 = l,/? = 0.

Means and Their Inequalities Theorem 31 [Room 200 1b] If η η η η (*ij, Pij > 0, 1 < г, j < η, and ^a»j = Σαν = Σ^' = Σ^' = 1' then 153 J=i J=i λ Ι Π Σ (ί1 ~ t)aij+tPij)*3 < *п(й), 1 < ί < 1. in particular 5η(α) < Ν Д ((1 - ί)α< + tan+1 - г) < On (α), 1 < ί < 1. D D These follow from I 4.2 Theorem 18 on taking f(x) = — log x. A simple question that arises from (GA) is whether or not it is always true that the geometric mean is nearer to the arithmetic mean than it is to the harmonic mean. In the case of η = 2 and equal weights the answer is immediate since Й(а, Ь) < 0(α, b) < Sl(a, Ь), and ϋ(α, b)Sl(a, Ь) = 02(a, 6) easily imply that (St(a, b)-0(a, b)) -(<8(a, Ь)-£(a, Ь)) = Sl(a, Ь)+й(а, &)-2^(Я(^Щм) > О, or χ < a(a)-g(a) < Я(а) 5(a)-Й(а) Я fe)' However if n > 2 it is possible for both 0n(a) to be nearer to 5ln(a) and for 0n(a) to be nearer to S)n(a); see [Scott]. The general situation is elucidated by the following theorem of Pearce & Pecaric; [Pearce Sz Pecaric 2001], see also [Lord]. Theorem 32 If а > 0 and η > 2 and α is a non-constant η-tuple then 1 ^ S£(o) - β°(α) w_ 1λ/2Ι„(α)Α« In particular 1 < 2ίη(α)-6η(α) < _ _2ln (a) (60) η - 1 ©„(a) - £n(a) £„(a) D The case η = 2 can be handled as above so assume that η > 3. ν (η—1)/η/^α/, ^1/n 0°(o)<(SG(fi)r X^nfe)) > by Sierpinski's inequality, (58), <u-is^(a) + ifl«(a), by(GA).

154 Chapter II This, on rearranging, gives the first inequality in (60). The second inequality in (60) follows by applying the first inequality to the η-tuple a-1. D Further generalizations of (GA) can be found in many places in this book, and of course turn up in the literature almost daily; see also [Alzer 1985, 1987b, 1989c, 1990c, 199Ы, 1992b; Chuan; Dragomir 1993, 1998; Hao 1990, 1993; Kritikos 1928; Sandor 1990a; Wang WL 1994a,b]. 4 Converse Inequalities (GA) in either of the forms 3.1(1) or (2) can be regarded as giving lower bounds to certain expressions. In general there are no interesting upper bounds unless the η-tuples α are restricted to a compact, see Notations 9. The topic of converse inequalities will be discussed fully for more general means in a later section; see IV 6. Here a few simple results are given as their proofs differ from those of the more general results. 4.1 Bounds for the Differences of the Means The basic result here is found in [Mond к Shisha 1967a,b; Shisha &; Mond]; see also [Alzer 1990b; Bullen 1979, 1980 1994b; Dostor; Tang] Theorem 1 Let a be a non-constant η-tuple with 0 < m < α < Μ, and гц and η-tuple with Wn = 1 ; then 0 < 2lnfew) - ®n(a;UL) < θ Μ + (1 - θ)τη - Memx~\ (1) where 0, 0 < θ < 1, is determined by 0(m, Μ; 1 - 0, 0) = £(rn, Μ). (2) There is equality on the right-hand inequality of (1) if and only if for some index set I, a subset of {1,... , n}, Wf = 6,ai = M,ieI,ai=m,i^L D We give two proofs of this result. (г) This theorem follows easily from proof (in) of 3.1 Theorem 1. The function / defined there has a unique turning point, a minimum, and so if as here, its domain is [m, M] the maximum of the function occurs at an end point, that is either at m or at M. Since this argument can be applied to an / that uses χ = a,i for any г, 1 < г < η, it follows that the maximum of the difference in (1) must occur when for all г, сц = т or αϊ = M. For such an η-tuple a, and some y, 0 < у < 1, %n(a\W.) - ®п{а;ш) = 2/M + (1 - y)m - Myw}~y = g(y), say.

Means and Their Inequalities 155 Simple calculations show that g has its maximum at у = #, where θ is given by (2), where £(ra, M) = (M — m)/(logM — logm), the logarithmic mean; see 2.4.5 Footnote 15, and 5.5 below. (гг) The method used in proof (Hi) of (J), I 4.2 Theorem 12 can be adapted to give an inductive proof. In the case η = 2 define the difference function as in the proof of 3.7 Theorem 23, D2(x,y;t) = D2(t) = Я(ж, у; 1 - t,t) - <б(х,у; 1 - ί,ί), 0 < ί < 1, 0 < χ < у; then D2(0) = Дг(1) = О and there is no loss in generality in the assumptions on ж,2/. Simple calculations give D'2(t) = (y - x) - (logy -\ogx)<3(x,y; \-t,t) and Dl2(t) = —(logy — log ж) <б2(х,у;1 — t,t). Hence D2 is strictly concave and so positive unless t = 0 or t = 1. Further D2 has a unique maximum at t = θ where Ό'2(θ) = 0, that is when <32(x,y; 1 - θ, Θ) = £(ж,2/), which is just (2) in this case. That is 0 < D2(t) < Ό2(θ), with equality on the left if and only if t = 1 or 0, and on the right if and only if t = Θ. This gives both a proof of (GA), and of the above result for η = 2. If we regard D2 as a function of χ then simple calculus show that it is a strictly decreasing function, while as a function of у it is strictly increasing. Hence if и < χ < у < ν, with not both χ = и and у — ν then D2(x,y;t) < D2(u, v;t). In particular as a function of t the maximum of D2(x,y-, t) is strictly less than the maximum of D2(u, v\t). This remark is used in the sequel. We now consider the case η = 3. Define, for 0 < χ < у < ζ, £>з(ж, у, ζ; s, t) = D3(s, t) = Sl3(z, 2/, z\ 1 - s - i, s, i) - 03(x, у, г; 1 - s - i, s, i), and 0 < s < 1, 0 < t < 1, 0 0<s + t<l; that is (s, i) lies in the triangle T of I 3.2 Theorem 12. Note that ifs = 0, t = 0ors + t= l then D3 reduces to a case of η = 2. There is no loss of generality in the restriction on the numbers x,y,z since if all are equal D$(s, t) = 0, and if two are equal D3 reduces to a case of η = 2. Since D is continuous it attains its maximum and minimum on T, and if this is an interior point we must have that — =(y -x)-&2 (ж, 2/, г; 1 - s - ί, s,) (log у - log x) = 0, —- =(z - ж) - 02 (я, 2/, г; 1 - s - *, 5, £) (log ζ - log x) = 0. So at such a turning points £(ж, у) = £(ж, г) ί = 02 (ж, 2У, 2;; 1 — s —i, s,) j. However £(x, 2/) < £(x, 2), see below 5.5 Theorem 10.

156 Chapter II Hence the maximum and minimum of D3 occur on the boundary of T; but on the boundary of T, as we have noted, D3 reduces to a D2, and we know that D2 is positive except at the end points where it is zero. Further the discussion of the η = 2 case shows that on each edge D3 has a local maximum and that the largest of the three of these occurs on the edge s = 0, since on that side the numbers χ. ζ are used. Further the maximum occurs when t = Θ, given by &2(x,y;l -θ,θ) = £(ж,2:). This proves that 0 < D3(x,y, z;s,t) < D2(x,z;l - θ,θ), proving (GA) in this case, and the above result. The method clearly extends by induction to any value of n. D The following results are of a different type. Theorem 2 Let a,w_ be as in the previous theorem, then: (a) [Cartwright L· Field] -4 71 -4 Π fc=i ,ГГ) (b) [Mercer A] fc=l fc=l 1 η 1 η fc=l fe=l Remark (i) The left-hand side in (a) with 2ln instead of <6n is the original result; the stronger form, that has a similar proof, is due to Alzer; [Alzer 1997a]. Remark (ii) The equal weight case of (a) has been improved; [Rasa]. Remark (iii) The result in (b) is not comparable with that in (a) and is found together with several other similar inequalities, in [Mercer A 1999)-. see also [Mercer A 2000, 2001, 2002]. The proof of (b) depends on the following interesting mean-value theorem due to Mercer. Lemma 3 Ifa.w_ are as in the previous theorem, η > 2, and if f.g £ C2(m.M) then for some y. m < у < Μ, an(/(a);a) -/(ЯпОкаО) _ f"{y) %г{д(а);ш)-д(Яп(а;ш)) д"(у)'

Means and Their Inequalities 157 provided the denominator on the left-hand side is never zero. Remark (iv) For an extension of this result and of Theorem 2(b) see III 6.4 Theorem 8. 4.2 Bounds for the Ratios of the Means In the equal weight case a very simple converse inequality has been given by Docev; the proof is based on the centroid method, see I 4.4 Theorem 24; [Mitrinovic & Vasic pp.43-44], [Docev]. Theorem 4 Let a, M and m be as in Theorem 1, μ = M/m, then ante) < (м-1)/^-1} () 0n(a) elog/i ' l j Further the right-hand side of (3) increases as a function of M, and decreases as a function ofm. D Since f(x) = — log ж, m < χ < Μ is strictly convex the centroid of the points (а^Дсц), 1 < i < n, the point (21п(й;ш)> ~ log^nfe W.))i lies strictly above the graph of / and below the chord t joining (m, f(m)) to (M, f(M)). If then 7 is chosen so that the graph of д(χ) = η -\- f(x) is tangent to t it is immediate that the centroid lies below the graph of g, that is — log0n(a;w;) < 7-log2ln(a;u0; or 2ί. (α) < e7. It remains then to calculate 7. Since i is a tangent to the graph of p, at (жо, Уо) say, 1 log^ 2A)-Hogm -log μ - 7 = ?A)-f-logXo, ~ жо m(/i — 1) xq — m πι(μ—1) These easily show that e7 is the right-hand side of (3). Further note that since / is strictly convex increasing Μ increases the slope of i, while decreasing m decreases the slope of i\ see I 4.1 (3). This observation readily implies the last part of the theorem. Π Remark (i) Inequality (3) is referred to as Docev's inequality. The following result also only considers the equal weight case; see [Loewner L· Mann]. Theorem 5 If 0 < а < —Ц— < β < 2, 1 < г < η, then SU (α) < /Mgh» < (β/α)[«(/*-0/(/>-°)]0-n+i -\<бп(а)) ~ 1 + (β-α) [η{β - I)/{β - α)] - (η - !)(/? - 1)'

158 Chapter II D We first find the minimum of f(x) = ΠΓ=ι(α + хг) m ^ne domain defined by η —α < —ρ < Xi < g, 1 < г < η; V^ a^ = c, ~^P < с < ng. If η = 2 the minimum is at (—p, с 4- ρ) if cp < g, or at (g, с — g) if с — g > —p. In either case, at most one of £1,2:2 is different from —ρ or g. We now show that this is true in general. Using Lagrange multipliers we se that the minimum occurs on the boundary and so one of the elements of ж, xn say, is equal to —ρ or q. Assume that xn~q. Now we have to find a point at which ΠΓ=ι (а+жг) nas its minimum in the domain n—1 -a < -p < х± < g, 1 < i < η - 1; ^Xi = c- q, ~(n - l)p < с < (η - l)g. The result now follows by an induction argument. Now apply this to the case a — 2ln(a), x% — &* — 2ln(&)> when с = 0. From the above 2ln(&) — ^п(й) is maximized if / is minimized, and that occurs if at least (n — 1) of the elements of χ are equal to either — p2ln(a) or to g2ln(a)· For simplicity put %i(Ql) = 1 and let ^ = —Ρ, 1 <i <r, and Жг = #57" + 1<2<7" + 55 r + s = η — 1. Then —p<xn = rp — sq < g, and so r < nq/{p+q) < r + 1. Now either nq/{p+q) is not an integer when r = [nq/(p + g)], or it is an integer when it is equal to either r or r + 1. In the latter case s = nq/(p + g), xn ρ so we could take r = nq/(p + q). In both cases therefore: r = [ng/(p + g)], s = n-l- [nq/{p + g)], жп = (ρ + q)[nq/(ρ + q)] -nq + q, and putting а = 1 — ρ, β = 1L -\- q simple rearrangements lead to the theorem. D Theorem 6 [Dragomir] Ifa,w_ are two η-tuples then fa - аз)2 max < -—7 f, ; f > < exp -—7 > WiWj- D The proof uses the following inequality proved in [Dragomir, Dragomir L· Pranesh; Dragomir & Goh 1996}; if Wn = 1, 0 - Σ ^ loS a« - loS ( Σ ™<α<) - Σ WiWj (^-Qj)2 i<i<j<n D A very simple converse inequality for the η = 2 case of (GA), in the form 2.2.2 (5), has been given by Zhuang, [Zhuang 1991].

Means and Their Inequalities 159 Theorem 7 If 0 < α < α < A, 0 < β <b < Β, ρ > 1 and p' the conjugate index then ap fcP' — + -T < Kab, (4) ρ ρ' ч where 'qP/p + BP'/p' АР J ρ Λ-βν' J ρ' К = max aB ' Αβ αΡ Ψ ι Π The function f(a,b) = ( 1 -) /ab, a > 0, b > 0, attains its minimum, ρ ρ' I namely 1, inside the rectangle in the, hypotheses, when ap = Ψ ; this is just the case η = 2 of (GA). Further this is the only turning point of /. So the maximum on the given rectangle is on the boundary. It is then clear that this is at either the point (a, B), or (Α, β). D A completely different result has recently been given by Alzer, [Alzer 2001a]. To state the theorem we need the following definitions. The digamma or psi function is Τ(χ)=φ(χ + 1) = (log ж!)'. The derivatives of this function are called the the multigamma, or polygamma, functions, oo 1 *Ю{х) = (-1)Ык\'£ z>o;fc = i,2,..·; see [DIpp. 71-72], where there are further references for these functions. Theorem 8 If α and w are η-tuples, η > 2, Wn = 1, and if к G W, then - Kk\(ot ( . \\» (5) with equality if and only if α is constant. The exponent on the left-hand side, k, cannot be replaced by any larger number. ϋ The proof of (5) depends on showing that the function /(*) = log (**№<*>(*:) |), z>0, is strictly convex, when the result follows from (J). For the rest if α is not constant and if (5) holds with exponent к replaced by а then, EL1^log|^^)(a,)|-log|^)(2in(a;^))| a < log %n (a; w) - log <3n (a; w)

160 Chapter II which, on letting αγ —► oo and using an asymptotic property of the multigamma function, log \ψ^(χ)\ ~ —/clogχ, χ —► oo, implies that a < k. D Remark (ii) For another converse inequality due to Alzer see below, 5.7 Theorem 18. 5 Some Miscellaneous Results In this section various properties of the elementary means of this chapter are discussed. The various results are not related to one another nor, in general, to the inequalities given earlier. 5.1 An Inductive Definition of the Arithmetic Mean We prove that the equal weighted arithmetic mean of η-tuples can be defined from equal weighted arithmetic means of (n — l)-tuples. This fact has been used extensively by Aumann in his study of the axiomatics of means; see [Aumann 1933a,b]. Theorem 1 Let α be an η-tuple and define the η-tuples g^r\ rGf, recursively as follows: gS1) = the arithmetic means of the η possible (n — l)-tuples from a; a(r) = the arithmetic means of the η possible (n — l)-tuples from a^r-1\ r > 2. Then limr_^oo a[r) = 2ln(a), 1 < i < n. D Assume for simplicity that for all г, а± < a^ · Then the result is immediate from the following identities. ΣΧ° =Έ^~1] =Έ«^> 2; αΜ - af> = °" ^ , г > 2. D Remark (i) This result has been extended; see [Kritikos 1949]. 5.2 An Invariance Property Given a sequence a it is natural to ask which of its properties are also properties of the sequences of means, 21, (3, defined in 3.4. We will only consider the sequence 21 in the equal weight case, and in several instances the answers are immediate: (a) if m < a < Μ then m < %_ < Μ since the arithmetic mean is internal, 1.1 Theorem 2 (I); (b) if limn^oo an~ a then, by a well known result of Cauchy, limn_^oo 2ln(a) = a, [Hardy p. 10}]

Means and Their Inequalities 161 (c) if a is increasing, strictly, so is 21, using internality of both the arithmetic mean and the weighted arithmetic mean and the formula η 1 2tn+i(a) = ^τγ^η(α) + ^γαη+ι· This last case is generalized in the following theorem. Theorem 2 (a) If α is k-convex so is 21· (b) if α is bounded and of bounded k-variation so is 21. D (a) Since α is /c-convex then Akan > 0, η e N*. Hence ϊ-< (i - 1)! l ~ or equivalently |^§д*вд > о, which implies the result. (b) By I 3.1 Theorem 3, a being bounded and of bounded k-variation implies that α = b — c, where Ь, с are /c-convex. So by (a) 2in (b) and 2in (c) are bounded and /c-convex. The result then follows since 2in(a) = 2ln(b) — 2ln(c). D Remark (i) Part (a) of Theorem is due to Ozeki. Later Vasic, Keckic, Lack- ovic & Mitrovic extended the case к = 2 to weighted means, finding necessary and sufficient conditions on the weights for the result to hold; see also [Andrica, Rasa L· Toader; Lackovic L· Simic; Mitrinovic, Lackovic L· Stankovic; Ozeki 1965; Popoviciu 1968; Toader 1983; Vasic, Keckic, Lackovic L· Mitrovic]. The following slightly different result is due to Lupa§ and Sivakumar; [Lupas 1988]. Theorem 3 If α is a sequence such that mp < Apan < Mp, η = 0,1, 2,... then -^-<A^n(a)<-^-n = 0,l,2,.... p+1 p+1 5.3 Cebisev'S Inequality We now consider the problem mentioned in 1.1 Remark (vi): to obtain a relation between 2in (a), 2in (b) and 2in(ab). Theorem 4 (a) If a, b and w_ are η-tuples with α and b similarly ordered then, Яп(а;ш)М&ш) < %n(ab;w), (1)

162 Chapter II If a and b are oppositely ordered then (~1) holds. In both cases there is equality if and only if either a, or b, is constant.. D (г) Suppose then that α and b are similarly ordered18. η η η i=i i=i i=i η η = 2_. (WiWjCLjbj — WiWjCLibj) = У. (^iwj^i^i ~~ WiWjCLjbi) i,j=i ij=i 1 n =— 2_. {^iWjajbj — WiWjdibj +WiWjdibi — WiWjdjbi) 1 2 У^ WiWj((li - CLj)(bi - bj) > 0, (2) which is equivalent to (1). Following I 3.3 Remark (viii) we can assume that both a and b are decreasing. Then since (2) contains the term wiwn(ai — an)(bi — bn) we see that the sum can be zero if and only if either a or b is constant. (w) A simple proof of the equal weight case of (1) can be based on I 3.3 Theorem 16. Let bU) = (b[j)..., blj)) = (fy, bi+1,..., bn, bu ... bj^), 1 < j < n, putting bo = bn. Since a and b are similarly ordered, Σ™=1 α,φί > Σ7=ι αΐ°ΐ\ 1 < 3 ^ η· Adding over j of these leads immediately to the required result. A similar proof can be given for (~ 1), when a and b are oppositely ordered. D Inequality (1), in the equal weight case, is due to Cebisev, and is called Cebisev's inequality; see [AI pp.36-37; DI pp.50-5; Hermite; HLP pp.43-44; PPT pp. 197- 198], [Herman, Kucera & Simsa pp.148-150, 159], [Cebisev 1883; Daykin; Djokovic 1964; Jensen 1888]. A history of this inequality can be found in the excellent expository paper [Mitrinovic L· Vasic 1974]. Generalizations are given in many places; see for instance [Pearce, Pecaric L· Sunde; Pecaric 1985b; Pecaric & Dragomir 1990; Popoviciu 1959a; Toader 1996]. Remark (i) The requirement that the η-tuples be similarly ordered is sufficient for (1) but it is not necessary; other sufficient conditions can be found in [Labutin 1947]. The problem of giving necessary and sufficient conditions for the validity of Cebisev's inequality has been solved; see [Sasser L· Slater]. Nanjundiah has given a proof of (1) that yields a Rado type extension; [Bullen 1996b]. It is based on the use of Nanjundiah's inverse arithmetic means, see 3.4.

Means and Their Inequalities 163 Lemma 5 With the notation of 3.4 Lemma 10 and assuming that η > 1 and (αη_ι, an), (bn_i, bn) are similarly ordered witJh equality only if an = an_i, or bn = bn_i. D This is an immediate consequence of the elementary computation D Theorem 6 If η > 1 and if α and b are both decreasing then Wn(2in(a Ь; ш) - %nfe ^)^n(& ш) >\¥п-г (Яп-1(аЬ;ш) - ^-1(а;ш)Яп-1(&ш)), (3) with equality only if an = 2ln_i(a;w) or bn = 2ln_i(b;w). D Since a and Ь are decreasing so are 2l(a; w) and 2t(b; w), 5.2 (c). So we can apply Lemma 5 using these sequences to get %пЧШшш)ЩЬ;&);ш)<К-1(Шш);ш)^1(Шш);ш) =^п1ШаЫш);ш) by 3.4 Lemma 10 (a); which is just (3). The case of equality follows from that of Lemma 5. D A weaker result in the equal weight case, due to Janic, is that n2(2in(ab) — 2tn(a)2ln®) increases with n; [A I p.206], [Djokovic 1964]. The best result in this direction is the following, due to Alzer; [Alzerl989e]. Theorem 7 If a, b are increasing η-tuples and w_ another η-tuple and к an integer with 2 < к < η, and α\ < α,, bi < bk then W2 W? ш _n (On{ah шУ&п(α; w)Kn(Ь; w)) > _fe (21, (a b; w)^lk(a; w)9Lk(b; w)). Remark (ii) The equal weight case of this result improves the quoted result of Janic and the result of Nanjunduiah as it says that (n2/{n — l)(2in(a)2in(b) — Sln(ab)) decreases with n.

164 Chapter II Example (i) Suppose that 0,2 = · · · = an = b<i — · · · = an = 1 then %n(Q b; w) -%n (a; w)%q(fe щ) 2tfc (u& ш) -2lfc fe ш)Я* (& ш) = WnWfc2(wiaibi + Wn - wi) - ^2(ΐϋιαι + Wn - w^A + Wn - wi) WfcW2(iuiaibi + Wfc - ил) - W2(tuiai + Wfc - iui)(iuibi + Wfc - ил) ' If now we let first a\ —> 1, and then let bi —> 1 the right-hand side has limit Wi(Wn — wi) ттД/ттГ г· This shows that the constant in the inequality in Theorem 7 W^Wk-W!) cannot be improved. Remark (iii) McLaughlin & Metcalf have studied inequality (1) from the point of view of functions of an index set; see 3.2.2. They showed that under suitable conditions the difference between the right-and left-hand sides, considered as a function of index sets, is super-additive; [McLaughlin L· Metcalf 1968b]. Another result due to Alzer is the following lower bound for the difference between the two sides in (1); [Alzer 1992c] Theorem 8 If α and b are strictly increasing η-tuples and w_ another n-tuple, η > 2, then ^η(αέ;^)-^η(α;^)^η(έ;^)>(21η(^2)-2ίη(^)) min {(а{-а^г)((^-bj^)}. This inequality is strict unless for some positive α, β, a^ = а\ + (г — l)a, and bi = h + (г - 1)β, 1 < г < η. D Let cn = min<i}j<n {(щ - ^-^((bj - Ьу-ι)}. Then if г > j, г г {di- a,j)(bi -bj) = 2J (a*; - afc-i) z2 (Prn-bm-i) k=j+i m=j + i г — z2 (ak -ak-l)(bm - bm-l) k,m=j+i i > Σ cn = (i-j)2cn. k,m=j + i Now if Sn denotes the left-hand side of the identity (2), the above calculation shows that i,3 = i 2 Simple manipulations show that this is just the inequality to be proved. D

Means and Their Inequalities 165 Remark (iv) A related result can be found in [AI pp. 340-341]· Writing (1) in the form η η η г=1 i=i i=i and applying this to infinite sums leads to various interesting elementary inequalities. For instance, [DIp.251]: tan χ tan у < tan 1 tan xy if 0 < ж, у < 1, or 1 < ж, у < π/2; arcsin χ arcsin у < - arcsin xy, if 0 < χ, у < 1. Remark (ν) For an interesting application of Cebisev's inequality see below 5.5 Remark (vii). The following definition allows us to generalize inequality (1). Definition 9 Give two sequences a and w_ we say that α is monotonic in the mean, increasing in the mean, respectively decreasing in the mean if the sequence 21ι(&; w), ^bfe w)·... is monotonic, increasing, respectively decreasing. Theorem 10 If the m sequences g^k\ 1 < к < т are monotonic in the mean in the same sense then m m k=l k=l Remark (vi) In the case m = 2 this is due to Burkill & Mirsky, and a simple proof of the general case has been given by Vasic L· Pecaric; see [Burkill Sz Mirsky; Vasic L· Pecaric 1982a]. For further generalizations see [Daykin; Pecaric 1984c; Sun X H; Vasic L· Pecaric 1982a]. Remark (vii) The concept of monotonic in the mean has been extended to sequences k-convex in the mean; see [Toader 1983, 1985]. 5.4 A Result of Diananda Theorem 11 If α and w_ are η-tuples, к G N* and a>k, then

166 Chapter II In particular if α > η — 1 then 1 < R < e. Further : (a) R = 1 if and only if all the elements of α are integers, and (b) R tends to (1 + - ) if and only if each element of α tends to (k + 1) from below. Remark (i) This result of Diananda has been generalized by Kalajdic; [Dianada 1975, Kalajdzic 1973). 5.5 Intercalated Means Let α < b and as in proof (г'г) of 2.2.2 Lemma 4 let us insert η arithmetic, η geometric and η harmonic means between α and b\ that is define the η-tuples а, д and h as follows: 6V/(n+1) , аЪ щ = аЛ -(b-α); 9i = а [ - ) ; /ι» = ; ; 1 < г < п. 71+1 W Ь_ ^ (6_α) The following facts are easily verified: 21η(α)=21(α,δ); &п(д) = &{а, δ); f)n(h) = f}(a,b); τ / \ τ «г / \ b — a lim S)n{a) = lim 2ln(p) lim S)n(g) = lim 2tn(ft) = log Ь — log а' log Ь — log a n—>oo ' b — a l/(b-a) у u l/(b-a) /ba\>-/{0-a) /ab\ lim (5n(a) = e"1 -^ ; lim 0n(ft) = e - The nine possible relations between these means are given in the following theorem. Theorem 12 On (a) > 0n(a) > Йп(а) > Mg) > 6n(fl) > in(g) > МЮ > <3n(h) > S)n(h). D Because of (GA) we need only prove that S)n(a) > 2tn(p) and S)n(g) > 2ln(h), and since the discussions are similar we just give a proof of the first. Consider then 1 η 1 η An& ="Eft = ή ΣβχΡ (losa + ^р[(1о%ь ~ 1οεα)) г=1 г=1 ι /•bgb logb-loga7ioga Ь — a log Ь — log a

Means and Their Inequalities 167 Similarly, since S)n{a) = (δΐ^α-1)) , 1-2 (7), consider ™ / in lA -i 1 /"b ι , log b-log α п^г b-aja b-a again using I 4.1 (5). This completes the proof of the first inequality above and in fact proves more, namely Йп(й) > ]—г^т > An{9)· log о - log a - The similar proof of the second inequality also proves more, namely log b-log а ЙпЫ > г > An(h). - о — a Π Remark (i) In fact I 4.1 Remark (xiv) shows that $)n(a) and S)n(g) decrease with η while 2tn(p) and 2ln(/i) increase, to the limits given above. Corollary 13 l/(b-a a + b ^fb'V^0-^ b-a r- > e 37 > л_и Ί„„„ > Vah (4) 2 \ αα / log b — log a log b- log а /а^\1/(Ь-а) 2аЬ ь-1 ν^α7 а + ^' Remark (ii) Nanjundiah has pointed out that this last inequality is a considerable improvement on the standard estimates for e and the logarithmic function, in particular I 2.2(9). This is an implicit in a later proof of the same inequality by Kralik which consists of substituting 1 + χ = b/a in I 2.2(10). [Nanjundiah 1946; Kralik]. /hb\1^b~a^ b — a The quantities e-1 ( — ) and -— are called respectively the iden- \aa J log b — log a trie and logarithmic means19 of а, Ь, written 3(a, b) and £(a, b); the definitions are completed by putting 3(a, a) = £(a, a) = a. Theorem 14 The means £(a, b) and 3(a, b) are strictly increasing as functions of both α and b. D This property of the logarithmic mean is a consequence of the strict concavity of the logarithmic function and I 4.1 Remark(v). The same result and the strict convexity of the function ж log ж, see I 4.1 Example(i), gives the property for the identric mean. D 19 The logarithmic mean has occurred earlier in 2 4 5 Footnote 15 and 4 1.

168 Chapter II Theorem 15 If a,b are positive numbers then min{a, b} < £(a, b) < 3(a, b) < max{a, b}; (5) more precisely 0(a, b) < £(a, b) < 3(a, b) < Sl(a, b). (6) All these inequalities are strict unless а = b. D (г) The outer inequalities in (5), the strict internality of the identric and logarithmic means, follow from the proof of Theorem 14. Alternatively the results are immediate from the mean-value theorem of differentiation, I 2.1 Footnote 1. Assume that а < b then: cv h\ /bg b-log αχ-1 £(α, b) = ( ) = с, lor some с, а < с < b; \ b — α J and τ ~/ 7 4 blogb-aloga log J(a, b) = —1-\ = log c, lor some c, a < с < b. (ii) The rest of (5) and (6) is proved in (4). A different proof of the inner inequality of (5) can be found in VI 2.1.1 Theorem 3; see also [Zaiming—Perez P]. We now give several proofs of the inequality for the logarithmic mean in (6), <б(а,Ъ) <£(а,Ь) <21(а,Ь), (7) and assume without loss in generality that а < b. (Hi) (7) follows from the left inequalities in I 2.2(14) by putting χ = — log(b/a), or by I 2.2(10) putting χ = b/α. (ii) Consider the graph of f(x) = ex, log α < χ < logb. Then, using the convexity of the exponential function and the Hadamard-Hermite inequality, I 4.1(4), we easily obtain the following /loga + logb\ n _ _ ч flogb хл exp log b 4- exp log α /Ί Ί Ί . exP (о (bgb-loga)< / exdx< —±L_EL_(iogb_loga)5 V Z J ./log a Z which is just (7). (Hi) Comparing the area under the curve у = x~l■, α < χ < Ь, with the area bounded by the lines χ = α, χ = b, the tangent to this curve at χ = (а + Ь)/2, у = 2/(а + b) and the X-axis gives the right-hand inequality in (7). Comparing the area under the curve у = ж-1, л/а < χ < yb with the area bounded by the lines χ = α,χ = b, the chord to this and the X-axis gives the left-hand inequality in (7)·

Means and Their Inequalities 169 (iv) If t >0 then by (GA), t2 + (a + b)t + ( °^- J > t2 + (a + b)i + ab > t2 + 2Vabt + ab, and so /»00 -| /»00 -ι /»00 -ι I (t + (a + b)/2ydt<J0 (t + a)(t + b))dt<J0 (t + Vab)***' which is just (7). D Remark (iii) Proofs (Hi) and (iv) are by Burk, [Burk 1985, 1987]] proof (iv)) is due to Carlson, [Carlson 1972b]. Other proofs of parts of (6) have been given, see for instance [College Math. J., 14 (1983), 353-356], [Perez Marco; Yang Υ 1987]. Remark (iv) In particular the above results show that these means are internal, monotonic and it is easily seen that they are also homogeneous. Remark (v) The result £(а,Ь) < Ща,Ь),а =£ b, occurs, heavily disguised, in [Polya & Szego 1951 p.9 (5)]. In the same reference an even stronger inequality is given 2 1 £(a, b) < -<3(a, b) + -Sl(a, b) < Ща, b), a^b; ό ό it is also heavily disguised; [Polya & Szego 1951 p.9 (6)]. A proof is given in VI 2.1.3 Remark (v). Sandor, [Sandorl995a], has shown that if α φ b then: -Ща,Ъ) <3(а,Ъ) <Ща,Ъ). e Remark (vi) The inequality used in proof (г), I 2.2(14), can be extended to sinhrr . . . sinh2x < tannrr < χ < sinn ж < —-—, χ > 0, V sinh χ + cosh χ when the same substitution, χ = - log(6/a), leads to S)(a, b) < ®(a, b) < £(a, b) < Ща, b) < Q(a, b), where jQ(a, b) = \/(a2 + b2)/2, the quadratic mean of a,b; see III 1. If in 1.3.1 Figure 1 the trapezium is divided into two equal areas by a line parallel to AB then its length is Q(a, b).

170 Chapter II Remark (vii) Proof (гг) of (7) gives another proof of (GA). Alzer has used the Cebisev inequality, 5.3, together with the logarithmic mean to give another proof of (GA); [Alzer 1991a). D Let a,w_ be n-tuples with a increasing and Wn = 1, then, putting 21 = 2ln(a;w), 6 = 0n(a;w), log^/2t) =-£кЖ' -г-п· Adding these leads to =έ^-(έ»Λ)(Σί(^)- w By hypothesis a is increasing and so by Theorem 14 the n-tuple (£(а$, 21), 1 < г < η) is also increasing. Hence by 5.3(1) the right-hand side of (8) is non-positive. This gives (GA), and the case of equality follows from that of 5.3(1). D Remark (viii) The logarithmic mean occurs in problems of heat flow; see [ Walker, Lewis & McAdams]. Dodd has pointed out that if a collection of incomes has a distribution between a and b that is proportional to their reciprocals then the mean income is £(a,6); [Dodd 1941b] Remark (ix) The topic of the identric and logarithmic means is taken up in more detail later, see VI 2.1.1 5.6 Zeros of a Polynomial and Its Derivative Theorem 16 Let ж be the η-tuple of the real distinct zeros of a real polynomial of degree η arranged in increasing order, and let у be the (n — l)-tuple of the real distinct zeros of its derivative, also in increasing order, then Щ(х) >Щ-\{у)\ 2lj(ccn_j+i,...,a;n) > 2lj-i(2/n-j+i, · · · ,2/n-i), 2<3<n, Further if f is a convex function on [x\, xn] and if A, 5 are denned by Ai = Οη-ι(£·), Bj = 21п-2(^) 1 < г < η, 1 < j < η - 1, then M/fe)) ^ *η-ι(/(]/)); Sin(/(4)) < Sln-i (/(£)). Remark (i) See [AI pp.233-234], [Bray; Popoviciu 1944; Toda]; see also V 2 Remark (x), and for another amusing result involving polynomials see [Klamkin L· Grosswald]. 5.7 Nanson's Inequality

Means and Their Inequalities 171 Theorem 17 If a is a convex (2n + l)-tuple and if b = {α2,α4,... , α2η}, с = {αι,α3,... ,α2η+ι} then Sin® <Sln+1(c) (9) wit-h equality if and only if α is an arithmetic progression. D Since α is convex we have that Δ2αη > 0, I 3.1. So in particular for к = 1,2,...,n k{n-k+ l)(a2fc_i - 2a2fc + a2fc+1) > 0, fc(n - fc)(a2fc - 2a2fc+i + a2fc+2) > 0. Adding these inequalities gives (9). The case of equality follows since Δ2αη = 0 implies that α is an arithmetic sequence. D Remark (i) An equivalent result has been generalized by Adamovic h Pecaric; see [AI pp.205-206; PPT pp.247-251), [Adamovic L· Pecaric; Andrica, Ra§a Sz Toader; Milovanovic, Pecaric L· Toader; Nanson; Steinig]. Another result involving convex η-tuples has been given by Alzer, [Alzer 1990d] Theorem 18 If α is an η-tuple with the (n + l)-tuple 0, αϊ,..., αη concave then *η(α) < |βη(α), and the constant is best possible. 5.8 The Pseudo Arithmetic means and Pseudo Geometric Means If a, w_ are η-tuples then W1 n Wri/Wl , ^ η 1 V^ / \ al (лсл\ an{a;w) = —a1-~-}^wiai, Qn(a;w) = — ^j^; (10) 1 X i=2 lli=2 ai are called the pseudo arithmetic mean, and pseudo geometric mean of α with weight w, respectively. Other forms of (10) are worth noting: On few) = αϊ ^n-i^wj, Wn/wi (11) vn\-i—J , 4\(Wn_ι)/ωι ' а1=Оп(а|,;ш) = вп(аь;ш); (12)

172 Chapter II where g$ and ab are defined as: Оп(а;ш), ifi = l, ab fSnfew), ifi = l, a^, if 2 < г < тг, г \ a^, if 2 < i < n. Remark (i) Simple examples show that in general the quantities defined in (10) do not satisfy internality, see 1.1. This explains the term pseudo mean. Indeed if we assume that a\ = mina, then Оп(д;ш) > mina = a\ implies, using (11), that ai > &n-i(&i; w'i)' which is false. Remark (ii) This lack of internality is not surprising since these pseudo means are cases of the arithmetic and geometric means with general weights that do not satisfy 3.7 (49), a condition that is necessary and sufficient for internality, see I 4.3 Remark(v); precisely, On(a;w) = 2lnfe w')> &п(я^ш) = ®п(а',ш'), where (Wn if г = 1, / I w\ W; = < l I w<j if 2<г <п. ν w\ Theorem 19 Ifa,ui are η-tuples then an(a;w) < Qn(a;w), (13) with equality if and only if α is constant. D We give several proofs of this result, (г) From (11) we have: M«; w) <а^,т (01п_1(а'1;ш1)Г(И'"~1)Л1\ ЬУ (~GA). 3·7 Theorem 23, ^ Wn/wi (^ ( / / \\-{Wn~i)/wi , //ΊΑν <a^ (en-iteiJWi)) , by (GA), (ii) By (12) and (GA), а!-21п(ай;ш) ><3п(ай;ш) η / { ^Wl/Wn Τ-Γ Wi/Wn this, on simplification is just the required inequality (Hi) Simple calculations, from (12), give Wn(pin(ab]w) - <5п(аЬ;ш)) = wi(0nfe ™) - ап(а;ш))· (14)

Means and Their Inequalities 173 This implies the result by (GA). (iv) Since (13) is trivial if an(a;w) < 0 we can assume it to be positive and apply the reverse Jensen inequality, I 4.4 Theorem 21, in a manner similar to the standard use of (J) to prove (GA), see 2.4.2 ρτοοϊ(χνί). In all proofs the case of equality is immediate. D Remark (iii) The equal weight case of (13) is due to Iwamoto, Tomkins & Wang but a full discussion of these pseudo means was given later by Alzer; [DI p.226], [Alzer 1990q; Iwamoto, Tomkins L· Wang 1986a]. Remark (iv) Inequality (13) as proof (iv) makes obvious is an example of a reverse inequality, another similar example of which is the Aczel-Lorentz inequality, see III 2.5.7. Remark (v) In the case η = 2 (13) is just a case of 3.7 Theorem 23. Further when η = 3 the result follows from the discussion in the proof (гг) of I 4.3 Theorem 20. It is shown there that the 0-level curve of D3(s, t) at the origin lies in the second and fourth quadrants. Hence the region {(s, t) : s < 0, and£ < 0} lies outside this level curve which implies that for such choices of s and t, (~J) holds. Corollary 20 With the same conditions as in Theorem 19, 0nfe^) - αη(α;^) > дп_г(а;ш) - а^Да; w); 8п(а;ш) > Ап-1(о;щ) An few) ~~ ап-Да;^)' with equality, in both cases, if and only if a\ = an. D The first inequality is an immediate consequence of (14) and (R). The second inequality follows in a similar manner from (P) and the following analogue of (14): \®n(afi;w)) \an(a;w)) The cases of equality follow from those of 3.1 Theorem 1. D Remark (vi) Clearly Corollary 20 gives Rado-Popoviciu type inequalities for these pseudo means that are in a certain sense sharper than the original (R) and (P). Remark (vii) As for (R) and (P) Corollary 20 implies the original inequality (13) and in fact can give better lower bounds, using the same idea as in 3.1 Corollary 3. In particular in the case of equal weights we get: &пШ ~ 4n(a) > max ; ——^ > max 2<i<n a\ ' on(a) 2<i<n a\ — (a\ — a^)2

174 Chapter II 5.9 An Inequality Due to Mercer Theorem 21 If a is a non-constant η-tuple such that m < α < Μ then M + m-2ln a;w > —-, r. D H0<m<t<M then obviously (t - m)(M - t) > 0 with equality if and only if either t = m or t — M. Equivalently, see I 2.2(23): Μ -bra — t > , with equality if and only if either t = m or t = M. Put t = di, 1 < i < n, and then take the arithmetic mean of the left-hand sides and the geometric means of the right-hand sides and the result is immediate using (GA). D This result can be found in [Mercer A 2002] and the substitutions m \—> га-1, Μ κ-► Μ~γ,α ι—► α-1, lead to М + га-21п(а;;ш) > .Ш > —γ. (15) <3n(a;w) м-i + m-i _ (йп(а;ш))" For a generalization see III 6.4 Remark (x).

Ill THE POWER MEANS This chapter is devoted to the properties and inequalities of the classical generalization of the arithmetic, geometric and harmonic means, the power means. The inequalities obtained in the previous chapter are extended to this scale of means. In addition some results for sums of powers are obtained, the classical inequalities of Minkowski, Cauchy and Holder, and some generalization of these results. Various generalizations of the power mean family are also discussed. 1 Definitions and Simple Properties Definition 1 Let a,w_ be two η-tuples, r £ Ш, then the r-th power mean of a with weight w is [(^|>*)1/r. **·***> #4r] (a; w) = I 0n(a;y, ifr = 0, f1) max a, ifr = oo, ν mina, ifr = —oo. As in the previous chapter we will just write QJtj^ if the references are unambiguous, 9^n (й) will denote the equal weight case, ЯКУ* will be used if η = 2; see II 1.1 Conventions 1, 2, 3; and if I is an index set the notation fflf (a;w) is used in the manner of Notations 6 (xi), I 4.2, II 3.2.2. Since #41] = KnMn] = ®nMn1] = Йп the power means form a natural extension of these elementary means. The case r = 2 is often called the quadratic mean, of α with weight w_, and written £1п(й',ш)] the case η = 2 was introduced in II 5.5. In addition the power means are sometimes called Holder means. The following result shows that the r-th power mean, r £ R, is a mean in the sense of II 1.1 Theorem 2 and II 12, Theorem 6, and that the special definitions given for the cases r = 0, ±oo are reasonable. Theorem 2 (a)IfreR then Tl[^](a;w) has the properties (Co), (Ho), (Mo), (Re), (Sy*), (Sy) in the case of equal weights, and is strictly internal, mina < 9Я^(а;гу) < max a, (2) 175

176 Chapter III with equality if and only if α is constant. (b) \imm[;]{a;w) = &n{a;w). (c) lim 97ц^(а;гу) = max α; lim 971^(α;υ;) = mina. r—>oo r—> — oo D (a) Immediate (b) We give several proofs of this result, and in all of them we assume that r e R*, and without loss in generality that Wn = 1. (i)[Paasche] Hm loga^'feffi) = lim MEH^Q r—►() = lim ΣΓ=^Κα[ log аЛ> by 1Ήό ital>s Rul η = ^(w* bg а») = 2ln (log (а); w), which gives the result by II 1.2 (8). The use of l'Hopital's Rule1 is easily justified. («) 1 n log OHM (ο; ω) =- log (Σ Wial) 1 n =- log (1 + r Y^{wi log Oj) + 0{r2)), using the Taylor expansion of the exponential function. The result is now immediate by I 2.2 (9), and II 1.2 (8). (in) By the note in the proof of I 2.1 Corollary 4 a\ = 1 + b{, where hi = 0(r) as r -> 0, 1 < г < n; and by I 2.1(5) aWiT = (1 + Ьг)^ = 1 + г^Ь» + 0(r2), as r -> 0. Hence, &n(a;w) = /nr=1q+bow<V/r аякГ,(й;«г) νΣΓ-χ^σ + Μ/ (1 + 0(r2))1/r = 1 + 0(r), as r -» 0, L'Hopital's Rule is:if f,g are real differentiable functions denned on the interval ]a,b[, bounded or unbounded, with g' never zero and if, with c=a or c—6, either limx_»c /(a:)=limx_*c g(x)=0 or limx_c \g(x)\=oo, then limx_>c Д||=Итх_»с /Ц > provided the right-hand side limits exists, unite or infinite; [EM5 pp.407-408}

Means and Their Inequalities 177 from which (b) is immediate. (iv) Assume r > 0, then as in (гг), iogimLrIteM)=iiog(X>X) ^((Σ^Ο"1)- by I 2.2(9), η r ι n = ^2m— > ^Wilogai, asr -> 0, = \og®n(a]w). The opposite inequality follows from inequality (r;s) given below, 3.1.1. The case r < 0 is handled similarly. (c) Assume r e R+ and without loss in generality that max α = αη, then (^)1/Гап<Ш^(аЖ)<ап, which implies the first part of (c). A similar proof can be given for the other part of (c). A slightly different proof of the equal weight case is given below in 2.3. D Remark (i) It follows from the internality that if α is a strictly increasing sequence then so is dSv£(a\w),n e N*; see II 5.2 (c). Remark (ii) More details of the behaviour of Wl$ as r —► 0, ±oo can be found in [Gustin 1950; Shniad]. Remark (iii) Another proof of the equal weight case of (b) is given below, 3.1.1 Remark (xi). The following simple identity is often useful. If r, s e R*, t = s/r, b = ar,c= a* then m^(a;w) = (m$(b-w))t/s = (SWM(su;))1/f (3) in particular of course, taking r = s in the first identity, Vt](a;w) = (2in(b;,w))1/S. (4) Remark (iv) These extend the identities II 1.2 (7), (8).

178 Chapter III THEOREM 3 If a,u,y_ are η-tuples with α increasing and Un = Vn, and if for 2<k<n,0<Vn- Vk^ <Un- U^ <Un = Vn, then for r e R*, <rfe)<<r]feu). Π Assume 0 < r < oo then η η ^ща\ =αλυη + Χ)№-ι - г/п)А<_1 η η >aiVn + X)(V5_i - Κ)Δ<_Χ = Σν, — 1 D Remark (ν) In particular if 0 < v_ < и with Un = Vn this reduces to a result of Castellano; [Castellano 1948). The general result is due to Pecaric & Janic. 2 Sums of Powers Before obtaining some deeper properties of the power means we study sums of powers. 2.1 Holder's Inequality The following theorem, Holder's inequality, is basic to all studies of power means and has many other applications.2 THEOREM 1 [Holder's Inequality] If α and b are two η-tuples and ρ > 1 then Σ-α<(Σ«ΠΣΌ "■ да i—\ г—ι i=i Ifp<l,p^Q, then (~H) holds. In both cases there is equality if and only if D [Case 1; ρ > 1] We give seven proofs of this basic result, (г) Inequality (H) can be written as By (GA), in the form of II 2.2.2 (5), the left-hand side of this inequality is not greater than V -—^ + -—% =i + - = l 2 ρ used in Theorem 1 and elsewhere is the index conjugate to p\ see Notations 4

Means and Their Inequalities 179 This completes the proof of (H) and the case of equality follows from that for (GA). (Й) Put a = ί/ρ,β = 1/ρ',α = ο</(Σ"=1 <%У'Р, Ь = &;/(Σ?=1 Щ)1/р' in II 2.2.2(4). Then summing over г gives (H) as in (г). (iii) [Matkowski L· Ratz 1993] If h :]0,oo[i—► Ш is concave and if c, d are two n-tuples then (~J) is equivalent to η h(Cn/Dn) > -λ-Y^dMa/di); further if h is strictly concave then there is equality if and only if с ~ d; I 4.2 Theorem 12. Taking h(t) = £1//p, ρ > 1, this inequality is Q/p^/p'>Ectl/?41/p', which after a simple change of variable is (H). The case of equality is immediate. (iv) [Soloviov] Let 7(a) = ( Σ™=1 ap J then 7 is homogeneous and strictly convex by I 4.6 Example (vii). Further 7^) = vP~l ( Y^L· νϊ) ~* * ■ Taking ν = Ь1/(р/_1), and u = α the support inequality, I 4.6 (23), is just (H). (v) A proof using II 2.4.6 Lemma 20 can be given; [Sandor &z Szabo; Usakov]. Using the notation of that lemma take fi(x) = α\χγΙν -f Щ x~l/v, χ e Μ = R^, 1 < г < η. So Σηί=1 inf^M fi(x) = KZti aibii where K = (p7p)1/p' + (p/p,)1/p- Then This by the quoted lemma completes the proof together with the case of equality. (vi) There is an inductive proof in proof(zz) of Corollary 2 below. (vii) See also 3.1.1 Remark (viii). [Case 2; ρ < 0] Then 0 < p' < 1, see Notations 4; put r = —p/p' then r' = 1/p', and r > 1. Now put с = a~p ,d= ap ψ then, by (H), Et^idi < (ΣΓ=1<)1/Γ(ΣΓ=1^')1/Γ'> which reduces to (~H). [Case 3; 0 < ρ < 1] Apply the above argument to p' since now p' < 0. D Remark (i) If υ; is some η-tuple it can be written asw = w}/pw}/v so (H) can be given a weighted form η η .. / η ι / ' (Σ^) (Σ^Ο · (ί)

180 Chapter III Corollary 2 Let ri > 0, 1 < г \< m, and put l/pm = Y^LX V7"* and let 0>i = (au> · ■ · > агп), 1 <i <m, then η m ч l/pm m η -. , Σ (ΕΚ) ) ^Π(Σθ) . (2) with equality if and only if the η-tuples д£*, 1 < г < m, are pairwise dependent. D We give two proofs of this generalization of (H). (г) Proof (г) of Case 1 of Theorem 1 is easily extended to this more general situation. (ii) A proof of (2), and incidentally of (H), can be given by an induction argument. If η = 2 then (H) reduces to aibi +a2b2 < (a{ + ар2)1^(ь{ + ti£)1/p', (3) which can be proved by one of the methods used to prove (H) in Theorem 1. We can now proceed in one of two ways. (г) Assume that (H) has been proved for all integers less than n, and that ρ > 0,q>0 and 1/p + 1/q = 1/r. Then / \r/p /n_1 \r/q У^ arjbj <arnbrn + ( 2_\ aj ) ( 22i Щ) 'by the induction hypothesis, *(Е«0Г"(ЕЧ)"'. *<»■ j=i j=i This proves (H), which is (2) for general η and m = 2. So now assume that (2) has been proved for all η and all ra, 2 < m < k. Then, (Σ(Π«βΓ)"4Σ<;(Π«.,ΓΓ ^ (Σ °S) (Σ (Π a^j ) - bycase ^ =2 of (2), and (2) follows by the induction hypothesis. (ii) The order of the induction can be reversed. Inequality (3) gives (2) in the case m = η = 2; keeping η = 2 assume the result for all integers less than ra. Then by an induction on ra similar to that above and with Y^=11/n = 1 ra ra ra 1Ь+П^П«<+^)1/Г4· ω

Means and Their Inequalities 181 The rest of the induction follows easily, as also does the case of equality. D Remark (ii) If m = 2 (2) can be written as (Σ«Γϊ(έ"0"*(έ'·')""· μ i—i i—i г —ι where p, q > 0 and 1/p 4- 1/q = 1/r. It is easily seen that if either ρ < 0, or q < 0 and if r > 0 then (~5) holds, while if r > 0 (5) holds. Finally if all three of these parameters are negative then (~5) holds. This can be put in a symmetric form as follows. Let a,b,c be three η-tuples such that abc = e and suppose that 1/p -f l/q 4- 1/V = 0, with all but one of p, g, r are positive, then (Σ<)"(Σ<>0"(X4 ^ w г—ι г=1 г—ι if all but one of p, q, r are negative then (~6) holds. This result can be extended torn, m > 3, η-tuples with all but one of the exponents having the same sign; see [Aczel L· Beckenbach]. Remark (iii) Inequality (4) is of some independent interest. In particular it shows, using I 4.6 Theorem 38, that if ρ : Rn ь-» R is a homogeneous polynomial of degree m, and if /(a) = р{д}1ш) then / is concave, and of course homogeneous of degree 1; see V 6 Theorem 3. The extreme generality of (H) leads to many inequalities turning out to be special cases in impenetrable disguises, as the next result illustrates; also see below, 2.5.4 Remark (iii). Theorem 3 (a) If 0 < s < 1 then (H) is equivalent to l-s Σ°№-'< (Σ*) (Σ*)"· (7) г=1 with equality if and only if а ~ b. (b) [Liapunov's Inequality] If α and w_ are η-tuples and r > s > t > 0 then η . η _ η _. (Σ«**?)Γ~ <(Σ™*ίΒ0Γ~ί(Σ>*ίΒ<)β~· (8) i=i i=i г=1 (c) [Radon's Inequality] If ρ > 1 then (Σ3.ΛΓ "f""

182 Chapter III with equality if and only ifa~b. If ρ < 1 then (~9) holds. D (a) This is seen to be equivalent to (H) by simple change of variables. (b) In (1) substitute ap = x\bp' = xr,p = (r -t)/(r -s),whenp' = (r-t)/(s-t). (c) (H) implies (9) by a simple change of variable; [HLP p.61], [-Radon]. D Remark (iv) Liapunov's inequality is a particular case of an inequality between the Gini means, see below 5.2.1 Remark(iv); see also [AI p.54; DI p.157; MPF p. 101; PPT ρ Λ17], [Allasia 1974-1975; Giaccardi 1956; Liapunov]. Remark (v) It has been observed by Maligranda that (7) was first proved by Rogers by a use of (GA), [Maligranda 1998; Rogers]. As this gives Rogers a priority he has suggested that (H) should be called the Rogers' inequality, or at least the Rogers-Holder inequality. Other forms of (H) are the following. Theorem 4 (a) If ρ > 1 then , l/p (Σα0 = sup ^сцЬ», where the sup is over all b such that Σ™=1 Щ = 1; the sup is attained if and only if а? ~ЬР'. (b) If ρ > 1 and a,b are η-tuples with An = Bn = 1 then with equality if and only if а ~ b. Remark (vi) The first part of this theorem is an extremely useful way of considering Theorem 1 and is the basis of a method of proof called quasi-linearization, see [BBp.23; MPF pp.669-679]. Remark (vii) Many other inequalities can be considered this way; for instance (GA): assuming without loss in generality that Wn = 1, η 0п(а;г/;) = inf 2_\wiciai, where С = {с; 0п(с;:ш) = 1}. ~ г—ι Remark (viii) Proofs of (H) can be found in many places; see [AI p. 50; BB p. 19; HLP p.21], [Abou-Tair L· Sulaiman 2000; Avram L· Brown; Holder; Iwamoto; Liu

Means and Their Inequalities 183 & Wang; Lou; Matkowski 1991; Redheffer 1981; Vasic L· Keckic 1972; Wang С L 1977; You 1989a,b; Zorio]. 2.2 Cauchy's Inequality If ρ = 2, when of course p' = 2, (H) reduces to Π П 1 /9 n 1/9 Σ«α<(ςΟ (ΣΟ - (η г=1 г—ι г—ι known variously as Cauchy's inequality, the Cauchy-Schwarz inequality3, or the Cauchy-Schwarz-Bunyakovskii inequality, see [Bunyakovskii\; we will refer to it as (C). A discussion of the history of this inequality can be found in [Schreiber; Zhang, Bao &: Fu]; an exhaustive survey of (C) and related inequalities can be found in [Dragomir 2003]. Obviously any proof of (H) provides a proof of (C). However, the simple nature of (C) allows for various direct proofs that show that 2.1 Theorem 1 holds for all real η-tuples a,b when ρ = 2. For instance, either of the following identities implies (C): η η η η Y^{aiX + bif = χ2 Σ α? + 2χ Σ aibi + Σ] $ г—ι ί=ι г—ι г—ι (ΣΟ (ΣΟ - (ί>0 = \ Σ (**i - αΑ)2· г=1 г—ι г—ι *J —1 The second identity is known as the Lagrange identity. For other proofs see [1938a,b; Cannon; Dubeau 1990a,1991b; Eames; Sinnadurai 1963]. Remark (i) Of course (C) arises from (H) by taking p — p'. By taking r\ = · · · = rm = m in (2) we get a simple extension of (C); [Kim S]. η m η Σ^····α-,·<Π(Σθ1Λη· 3 = 1 г=1 j=i Theorem 5 (Η) and (С) are equivalent. D That (H) implies (C) is trivial and for the converse we show that (C) implies the extension of (H) in 2.1 Corollary 2, (2). The proof is in five steps and we assume in Corollary 2, without loss in generality, that pm = 1. 3 This is K.Schwarz.

184 Chapter III (г) т = 2 and τ γ — Г2 = 2: then Corollary 2 is just (C). (гг) m = 2μ, μ £ Ν*, τ\ — · · · = Τ2μ = 2μ: the proof of this case of (2) is by induction on μ, μ = 1 being (г). So suppose the result is known for integers k,l <k < μ. Then Σ!Παν = Σ^ (Π °ч) ( Π α^) j —ι г—ι г^2/х-1+1 (η 2μ_1 \ 1/2 / η ·2· ч ι/^ Σ(Π4)) (Σ( Π 4)) ,μο, j = i i—ι ' j = i г=г2^_1 + 1 2μ η j/2m < JJ ( 2^ a?J ) ' ^y ^he induction hypothesis. (iii) Now let m be any integer, r\ = · · · = rm = ra, and suppose 2μ > m. Define o^·, 1 < г < 2μ, 1 < j < η, as follows: (<V/ j if 1 < г < m, 1 < J < n, (ffili^) , ifm<i<2M<j<n. Then by (гг) η m η 2μ ΣΠα^=ΣΠα^ 2μ П -, /ομ 771 Π Λ ,, j=ri г=1 j—ι г—ι г=1 j = i г=1 j=i (w) Let m be any positive integer, r, G Q, 1 < i < m. Then for some integers μ, μι, 1 < г < m, we have r^ = μι/μ, 1 < г < т. Define &ц = α^, 1 < г < га, 1 < j < w> then applying (гг'г) to the {a^} gives the result in this case. (v) The general case of real r^, 1 < i < ra, follows by taking limits. It remains to consider the case of equality. Let 1/r^ = qi + Pi, qi Ε Q, 1 < г < ra,

Means and Their Inequalities 185 then η m η m Гг(Яг+Рг) ΣΕΚ=ΣΙΚ( j = i i=i j—i г—ι η πι η πι ρ -Е(П«Г/(г") (П<Г/Р") j=i г—ι г—ι η πι ^ η τη ρ ^(ΣΓΚΓ^") (ΣΠ<Γ/Ρ") "· by the above, πι η η πι ρ ^Π(Σα«Γ(ΣΠβίΓ/Ρη) "' by the above up to И, г—ι j —ι j —ι г—ι πι η πι η ^Π(Σ<ί)*Π(Σα«)'-ьу theabove- m η =Π(Σ<])' г=1 j = i г=1 3=ι πι η ι l/^г In using the above arguments up to (iv) we have strict inequality unless the n- tuples a^, 1 < г < га, are pairwise dependent. This gives the case of strict inequality in the general case, and Theorem 3 is proved. D Example (i) Consider the part of an ellipse χ = (acos#, bsin#), 0 < θ < π/2, where 0 < b < a. The distance, d, of the normal at the point χ from the origin is given by d = |ж.£|/|£|, where t = (—asin#, bcos9). Simple calculations give that d = (a2 - b2)/vVsin2#-f-b2cos2#. Now by (C) Va2 sin2 θ + b2 cos2 θ = )/(a2 sin2 θ + Ь2 cos2 0) (cos2 0 + sin2 0) > (a+ 6) sin 0 cos 0. with equality if and only if θ = 9q where tan#o — \/b/a. Hence the minimum distance of a normal to the ellipse from the origin is (a — b), and it occurs when θ = #o; [Klamkin L· McLenaghan]. Example (ii) (C) can be used to give a very simple proof of (HA) in the equivalent form II 2.4.3(12); see [Batinetu]. Σ^)(έ?) * (Σ(^)1/2(?)1/2)2 = ^2; 2.3 Power Sums For an η-tuple α write η «Sn(r.o) = 53 of, r 6 К, 7гп(г,а) = ^(г.а), г 6 R*;

186 Chapter III as usual reference to η and, or a, will be omitted if there is no resulting confusion. We use (H) to obtain some properties of these quantities. Since 1Z(r) = nl^rUJV^}{a) we have by 1 Theorem 2(c) that limr^007^(r) = max α and limr^_007^(r) = mina; or just note that, maxa < lZ(r) < nl'r max a, r > 0; nx'r mina < 1Z(r) < mina, r < 0. This also shows that the equal weight case of 1 Theorem 2(c) can be deduced from the limit values of ΊΖ{τ). From the remark in the proof of I 2.1 Corollary 4 we have S(r) = n-f-O(r), r —+ 0. Hence, \imr^o1Zn(r,a) = oo, if η > 1. The same result follows from the simple observations, nl'r minα < 1Z(r) < nl'r maxa, r > 0; nl^r maxa < !Z{r) < n1^ mina, r < 0. Lemma б (а) ΊΖ is strictly decreasing; that is ifr<s then TZ(s) = (f>s)l/S < (ΣοήΦ =K{r). (10) (b) Both ofS,lZ are log-convex. More precisely if 0 < λ < 1 and r,sGl then «S((l - X)r + \s) < S^-x\r)Sx(s); (11) with, if λ φ 0,1, equality in (11) if and only if α is constant; if r, s > 0 then n((l-\)r + \s) <n^-x)(r)nx(s). (11') D (a) Assume that r > s and that S(s) = 1. Then for all г, 1 < i < n, we have af < 1. So if s > 0, a\ > af, showing that 1 < <S(r), and so 1 < lZ{r)\ and if s < 0, a\ < af, showing that 1 > <S(r), and so 1 < !Z{r). This completes the proof in this special case so now assume that S(s) = σ, and put, for all г, hi — di/a1^ = bi/7Z(s), when <S(s; b) = 1. From the special case we then have that 7l{r\ b) > 1, which is just (10). (b) First we consider S ±atX)r+Xs =Σ«)(1-λ)«)λ * (Σ<)(1_λ)(Σ«Ολ- ЬУ 2.1 (7). The case of equality follows from that for 2.1 (7).

Means and Their Inequalities 187 Now we consider ΊΖ . If !Z{r) > 1 then S(r) > 1 and so since \oglZ{r) = r_1 log<S(r) the result follows from the convexity of log<S and f(r) = r-1, using I 4.1 Theorem 4(e). If !Z{r) = ρ < 1 choose p', 0 < p' < ρ and let a[ = a,i/p'', 1 < г < η, when 7Z(a',r) = p/p' > 1 and so 7Z(a',r) is log-convex. However log7£(a/,r) = \ogTZ(r) — log/97, so \og!Z(r) is convex. D Remark (i) The proof of (b) can be found in [Beckenbach 1946]. Remark (ii) If w. is an ττι-tuple with Wm = 1 a simple induction extends (11) and (ll7) to <S(%m(r;w),a) < 0m(<S(rba),...,<S(rm,a);w), (12) 7г(21т(г;^),а) < Фт(Щги а),..., ?г(гт, a);w). (127) Ursell has made some interesting observations about (11) which we will now discuss; see [Reznick; Ursell). Consider (11) in the special case obtained by putting r = 0 and Xs = p, S(p)<n1-^sSp/s(s). (13) or equivalently ΊΖ(ρ) < n^p~^!Z(s). Inequalities (10)—(13) are best possible in the sense that given n,r, s, A,p the constants cannot be improved. However (10) and (13) are best possible in a stronger sense. Given n,r, s,p, S(s) the values of S(r),S(p) are given by (10) and (13). In the case of (13) equality implies that (7^(r)/7^(s))rs^1-r^ is equal to n, in particular it is an integer. In other words, given n, r, s, λ and appropriate S(r) and <S(s), we can still have strict inequality in (13) unless lZ(r)/lZ{s) has a special value. The determination of the exact range of this left-hand side given the sums on the right-hand side is completely determined by Ursell; see also [Pales 1990b] and IV 7.2.2. Remark (iii) The sums, S and ΊΖ, can be considered with weights; that is we define S(r,a]w) = ΣΓ=ι WiCLi an<^ ^(r>^^0 = {Y17=iwiai) Г- ^ ls immediate that Lemma 6(b) remains valid for these more general functions. Lemma 6(a) is considered in [HLP p.29) and [Vasic L· Pecaric 1980a)] it is valid if w_ > e, while if Wn < 1 the opposite inequality holds; see 3.1.1 Remark (ii). See also below IV 2 Theorem 13. Remark (iv) If r > 1 then | Щг, а) - K(r, b) | < min Ι Σ^=1 \dk — bik ||5 where the minimum is taken over all permutations {г1?..., in} of {1,..., n}. This answers

188 Chapter III a question in [Mitrinovic Sz Adamovic]; some generalizations can be found in [Milovanovic L· Milovanovic 1978; Pecaric Sz Beesack 1986]. Because of (HA) and Lemma 6(a) the following strengthens (12'). Lemma 7 If Wn = 1 then ^(imfc;i),a) <вт(7г(п,а),...7г(гт,а);ш), (14) with equality if and only if either τι or α is constant. D Raise the left-hand side of (14) to the power S) = S)n(a]w_) when we obtain, П П П ft П / П ft ft\ Wj$)/rj Σ-Ρ-ΣΟΚΟ snfeh-V"*) · by (2), since Y^j=1 Wj5)/rj = 1, and this implies (14). The cases of equality follow from those of (2). D Remark (v) By considering the case of constant a when (14) reduces to equality, it is easily seen from (10) that (14) is best possible in the sense that йт(а;ш) cannot be replaced by a smaller number. Remark (vi) Inequality (14) is in [McLaughlin L· Metcalf 1968a], where it is also pointed out that this inequality is best possible in another sense. The right-hand side of (12) cannot be replaced by any function of the ΊΖ{τι), 1 < г < η, that is less than or equal to the geometric mean but which is not the geometric mean itself. Lemma 6(a) can be used to extend (H) and 2.1 (5) as follows. Corollary 8 (a) Ifp, q > 0 and 1/p 4- 1/q > 1 then Σ«λ<(Σ«0 (ΣΌ · (b) Ifp, q, r > 0 and 1/p + 1/q > 1/r then i/r , » χΐ/ρ/ * χι/9 (Σμ»>'Γ <(Σ«0 '"(ΣΟ' with equality if and only if 1/p + 1/q = 1/r and αρ ^ bq. D (a) Put λ = 1/p +l/q> 1 and r = Xp > p\ then r' = Xq > q so 1/r/ ™ л1/г' Σ>^(Σ°0 г(Еь0 .ьу (н), i=i i=i i=i

Means and Their Inequalities 189 (b) This part is either (5), or reduces to (a) by a simple change of variable. D Remark (vii) The properties of 7£(r), r < 0, and their applications to the various results given above are easily obtained; see [HLP p. 28), [Vasic Sz Pecaric 1979b]. Remark (viii) In his very interesting paper Reznick, [Reznick], has considered precise bounds for various products of power sums. Some of his simpler results are that for real η-tuples α η <S(a; l)<S(q; 3) ~8- <S(a;4) ~ ' <S(a; l)<S(a; 3) <^vfc + f + o(n-n <S(a;2) Remark (ix) Given an η-tuple α inequalities of the form η Σ°?Α?<κ{<*,β,η№+β; are considered in [Beesack 1969} ; in particular ifa>l,a-f-/?>l and α is positive then К = 1; see [AIp.282]. 2.4 Minkowski's Inequality A very important consequence of Holder's inequality, 2.1 Theorem 1, is Minkowski's inequality. THEOREM 9 [Minkowski's Inequality] Ifp>l then φ*+>,y) "pi(|>?)"4|: Ч)1"-. <«) if ρ < 1, ρ =£ 0, then (~M) holds. In both cases there is equality if and only if D We give five proofs of this result. (г) Consider the left-hand side of (M), and assume ρ > 1: η η η Σ(°<+ад* = Σ α·(α·+W"1+Σ w*+ад*-1 ί(Σ<Ό"'(Σ(«<+'<>'Γ"" + (Σ^)1/Ρ(Σ(«ί+^)(Ρ")/Ρ> by (Η); from which (M) follows. If ρ < 1, p^O then (~M) follows by a similar argument from (~H).

190 Chapter III The case of equality follows from that for 2.1 Theorem 1. (гг) This proof uses quasi-linearization, see 2.1 Remark (vi) and [BB p.26]. Assume that ρ > 1; the other case can be dealt with in a similar manner. By the quasi-linearization form of (H), 2.1 Theorem 4(a), if L = {c; Σ™=1 с? = 1}, / \ ι/ρ then (Y%=i(ai + bi)p) = sup£GL Y^=1(a{ + 6»)с». Hence П 1 /r> П П (5^(oi + bi)A ^sup^OiCj + sup^bjCi, by I 2.2 (15) = (Ea?)1/P + (Σ6?)17"' by 2>1 Theorem 4(a)· (ш) [Konig; Maligranda 1995} With ρ > 1, £ > 0 and а, Ь > 0 consider the function f(t) = t1-pap + (l-t)1~pVp. Simple calculations show that f'(t) = (l-p)(t~pap - (1 - t)'pbp), and that / has a minimum at t = a/(a -f b). In other words f(t) > f(a/(a 4- b)), with equality if and only if t = a/(a 4- b), that is (a 4- Ь)р < i1"^ 4- (1 - ί)ι~νΨ, (15) with equality if and only if t = a/(a 4- b). Now using (15) f>+hr < ς {tl~p< + а - tf-^щ) i=i i—i \1/P Hence the left-hand side is not greater than the minimum of the right-hand side and the result follows by another application of (15). The case of equality is easily obtained from that in (15). (iv) [Matkowski 8z Ratz 1993} Use the same method as in proof (Hi) of (H), 2.1 Theorem 1, but now take h(t) = (14- t1/p)p, ρ > 1. (v) [Soloviov} Let /(α) = ί Σ™=1 αΡ ) then / is strictly convex and homogeneous, see I 4.6 Example (vii). So (M) is just I 4.6 (19). D Remark (i) Of course basic properties of addition extends (M) in a trivial way to the case ρ = 1.

Means and Their Inequalities 191 An important use of (M) is the proof of the triangle inequality, (T). If ρ > 1 then, Φ loi - b^)1/P =QT|K -a) + (Ci - ы)\р)1/Р <(Σ(Κ-^! + |^-Μ|)ρ)1/Ρ, <(έ(|θί - c^)1/P + (£(|с, - ^\ή1/Ρ, by (M). This inequality holds of course when ρ = 1 being a property of the absolute value. So if pp(a,b) = [Σ™=1 \сц — bi\p ) , ρ > 1, then we have the triangle inequality, £p(α, &) < Pp (a, c) + Pp (c, 6), (T) the essential property for showing that pp is a metric on the space of η-tuples, or equivalently that ||a||p = (Σ™=1 af) , ρ > 1, is a norm, ||α + 6||ρ<||α||ρ + ||έ||ρ. (TN) Remark (ii) (M) was first used in the form (TN) by F. Riesz, and for this reason Minkowski's inequality is sometimes called the Minkowski-Riesz inequality. For a similar reason the Holder inequality is sometimes called the Holder-Riesz inequality when written in the form ||αέ||ι<||α||ρ||6||ρ/; see [MPF pp.473-513], [Maligranda 2001]. The following extension of (M) is analogous to the extension of (H) given in 2.1 Corollary 2. Corollary 10 Иа{ = (α^,..., ain), 1 < г < m, and ρ > 1 then (η m \ 1/p ran Λ , Σ(Σ-)*) *Σ(Σ·&Γ· («) with equality if and only if the η-tuples а{, 1 < i < m, are pairwise dependent. D Inequality (16), and so (M), can be proved by induction, as was the case for (H), see 2.1 Corollary 2 proof (ii). We start with the simplest case, the case m = η = 2 of (16): ((αϊ + hY + (α2 + Ъ2У)1/р < (of + 4Ϋ/ρ + (Ь? + Щ)1^.

192 Chapter III This can be obtained from 2.1(3), the case η = 2 of (H), in the same way that (M) was obtained from (H). Then, as with the inductive proof of 2.1 Corollary 2, we can either fix m = 2 and give an inductive proof of (16) for all n, and then for all m, or we can proceed the other way round; see [HLP p.38]. D Remark (iii) Much work has been done on this inequality; see for instance [Szilard; Zorio]. 2.5 Refinements of the Holder, Cauchy and Minkowski Inequalities The inequalities (H), (C) and (M) have been subjected to considerable investigation, resulting in many refinements; some of these are taken up in this section. 2.5.1 A Rado type Refinement The simplest proof of (H), or of 2.1 Corollary 2, depends on (GA), and it is natural to ask if it is possible to refine (H) by using (R); [Bullen 1974]. Theorem 11 With the hypotheses and notations of 2.1 Corollary 2 put πγ=1(ς;=χ·) then ifm>2 Sm(a) < Hm_!(a) Η , (17) Pm Pm — i Τ with equality if and only if for j — 1,. .., η Π Σ3=1 arnj \i=i Hj=i <V D From (R) μ/ν 1/Pn-1 ^ / m Ί 77i \ . / m — 1 T m — 1 \ Putting %'- -,j = l,...,n (18) and summing the resulting inequalities over j leads to Pm(l-Sm(a)) > pm-i(l -Hm_!(a)), which is equivalent to (17).

Means and Their Inequalities 193 The case of equality follows from that for (R) given in II 3.1 Theorem 1. D Remark (i) If m = 2 inequality (17) reduces to Нг(а) < 1, which is just (H). Remark (ii) Repeated application of (17) leads to Sm(a) < 1, which is just 2.1(2). Remark (iii) By terminating the process in (ii) one step earlier and allowing for rearrangements of a^·, together with similar rearrangements of r\ the result in (ii) can be improved as in II 3.1 Corollary 3. Remark (iv) For a Rado type extension of (M) see 3.2.6 Corollary 26. 2.5.2 Index Set Extensions Define the following functions on the index sets X: »«-(Ε<Γ(Σί)*-Ε·* iei iei iei μ/)=((Σ<)1/ρ+(Σ^)1/Ύ-Σκ+^. ^ iei iei J iei The inequalities (H) and (M) show that if ρ > 1 then χ, μ > 0. In fact the functions χ and μ are also increasing and super-additive as the following more precise theorem shows; [Everitt 1961; McLaughlin L· Metcalf 1967a, 1968b,1970]. Theorem 12 Ifp> 1, I, J eX then x(/)+x(J)<X(/UJ), (19) /x(J) + /x(J)</x(JUJ). (20) There is equality in (19) if and only if (J2ieI a?, Y.iej ai) ~ (Eie/6?'» Σ^ί), and in (20) if and only if &iei *l Eiei Щ) ~ (Σί€ j *?, Σί€ j Щ) · D We first consider (19). By 2.1 (3) а^Ъ^' + α)Ι%ψ' < (αϊ + ^''(h + b2)^\ with equality if and only if (01,02) ~ (bi, 62)· Substituting αϊ = Σ%εΐ аЬ а2 = E*GJaf' Ъ^ = Т^ге1Ъ1 > Ь2 = Еге j bf leaxis immediately to (19), and the case of equality. Now consider (20); by (~M), ((αϊ + h)1^ + (a2 + b2)1/p)P > (αϊ/ρ + α\'*)* + (b\/p + Ь^)*,

194 Chapter III with equality if and only if (αϊ, аг) ~ (bi, Ьг)· The result, and the case of equality, follow on putting αϊ = £-G/ a?, a2 = £iG/ Щ, Ьг = £iGJ a?, b2 = Σ^J *£· D Remark (i) In the paper quoted above Everitt pointed out that the index set function (Σαη1/ρ+(Σ^)1/ρ-(Σκ+^)1/ρ i£l iel iel associated with (M) is not monotonic in either sense. In the second paper mentioned above McLaughlin & Metcalf studied the function μ, as well as a certain ratio associated with (M); see also [Dragomir 1995a; Vasic L· Pecaric 1983]. 2.5.3 An Extension of Kober-Diananda Type With the notations of 2.1 Corollary 2 assume pm — 1 and define R — max1<i<rrir^, τ — min1<^<rn r,· and \ 2 1 in ι n «■■ΣΣ Στη η л уг^т п ι ' \ 2 ^ = όΣ Σ — %' / "μ г=1 \j,fc = i Г{Гк V ^г=1 αΰ' V 2^i=\ aik Theorem 13 With the above notations m η η m ί1 - S) Π(Σ<ί)1/Γί * Σ ΕΚ * ma4°- α -^»)Π(Σ^)1/Ρ<}5 ran η m ТуГ m η γ r lib ft II ΠΙ jy- \ ΤΤίΎ^ r-^i/rt- ^-^t-t .. r„ /. л„ ДГт-пДАЦ-'' v ' г=1 j = \ j = i i=i i=i j = i (1-^ιτ)Π(Σ^)1/^ΣΠ^^40'(1-τ1)Π(Σ<])1Μ}· D The proof is based on the method used to obtain 2.1 Corollary 2 and Theorem 1. Use inequalities II 3.5(34), (35) applied to a sequence b, make the identification 2.5.1(18), and proceed as in the proof of 2.1 Corollary 2. D Remark (i) The cases of equality are discussed by both Kober and Diananda; Diananda gives a similar refinement of (M); [Kober; Diananda 1963a,b]. 2.5.4 A Continuum of Extensions As has been remarked (C) is a particular case of (H). In fact (H) can be used to prove much more. Let aij, r^, pm be as in 2.1 Corollary 2; then we have the following result. Theorem 14 Ifre M* and w is an η-tuple define the function f as follows m η m /(-) = Π(Σ-^Γ/ΓΠ<Γ)1/η i=i j = i k=i

Means and Their Inequalities 195 Then f is log-convex, increasing on [0, oo[, and decreasing on ]— oo,0], strictly unless f is constant. D If the m-tuples are not proportional then by 2.3 Lemma 6(b) and 2.3 Remargin) η m fc=i j = i k=i k = i is log-convex, 1 < г < т. Hence, I 4.1 Theorem 4(d), bg f{x) = Σ —log gi(x), is convex. The rest of the proof is similar to that of I 4.2 Theorem 18; see [PPT pp.90,118], [Mitrinovic L· Pecaric 1987]. D Remark (i) Beside the paper just mentioned this result is discussed in [Daykin &: Eliezer 1968; Eliezer L· Daykin; Eliezer L· Mond; Flor; Godunova L· Cebaevskaya; Mon, Sheu & Wang 1992b]; see also IV 5 Example (xiv). Remark (ii) The most interesting case of Theorem 13 occurs on putting г = pm when /(0) is the left-hand side of a weighted form of 2.1(2), while f{pm) is the right-hand side. Since /(0) < f(x) < /(pm), 0 < χ < pm, we get a continuum of generalizations of 2.1(2), and so of (H): if 0 < χ < pm then / η m ч 1/ pm πι η πι πι η 1 , (Σ^(Πή,-Γ) *П(Е^""" П"£Г)1/р<*П (Σ-,4θ \ j = i i=i ' i=i j = i k=i i=i 3 = 1 In this case the function / is a constant if the η-tuples a[*, 1 < г < т are proportional. Remark (Hi) The particular case, m = 2, η = Г2 = 2, of the result in the previous remark is in [Callebaut] and is sometimes called CallebauVs inequality, [PPT p. 118]: it0<x<y<l then (E агЬг)2 < £ αί+·6ί-έ αί-6ί+· < jl· α]^^± ^^ <^±Ъ] г=1 г=1 г=1 г=1 г=1 г=1 г=1 This result is an illustration of the comment that precedes 2.1 Theorem 3. Callebaut proved one half of his inequality using (H) and remarked that the other half was not apparently deducible from (H), and so gave another proof. However a simple proof using (H) was given almost immediately, [McLaughlin L· Metcalf 1967b]; see also [Steiger].

196 Chapter III Theorem 15 If 0 < ж < 1 then ,f 1=1 i,j = l η η η η < (j>? + 2x Σ α^) (Σ Ь< + 2χ Σ b*i) D Using the identities Σ^-=ι aibj = Σ?=ι αΐ Σ?=ι ^ -Σ?=ι α^> 2Σ?=) a^j = (Σ?=ι α0 -Σ?=ι α?> and 2£)7Li b^· = (X)^=1bi) - Σ^=1^, and putting у = 1 - χ, (21) can be i<4 written as η η (ζΣ^Σ^+^Σ^)2^ («(^α»)2+ϊ/Σα«2)(2;(ΣΜ2+2/Σδ«2)· г=1 i=i i=i i=i i=i i=i i=i On multiplying this out we obtain the equivalent inequality η η η η η *κΣ&Σ* - α> ΣΜ2+2/2((Σα')(Σδ2) - (Σ«α)2) > ο, j=i i=i i=i i=i i=i i=i which is an immediate consequence of (C). D Remark (iv) If χ = 0 then (21) reduces to (C). Remark (v) This result is due to S.S. Wagner, but the above simple proof is by Flor; [MPF p.85], [Andreescu, Andrica &: Drimbe; Flor; Wagner S]. 2.5.5 Beckenbach's Inequalities The first part of 2.1 Theorem 1 can be interpreted as follows. Given αϊ and b then (H) holds for all CL2,..., αη, with equality if and only if аргЬрг' = ajb?' = а?Ь?\ 2 < г < η. This has been generalized as follows in [Beckenbach 1966]. Theorem 16 Let α and b be η-tuples, ρ > 1, and 1 < m < n; define the n-tuple α as follows: di, if 1 < г < m, 0>i = < ^mJ г / 1 , ifm<i<n; Σ,·=ι азЬэ

Means and Their Inequalities 197 then given αγ,..., am and b, (ΣΓ=Χ)1/Ρ > (ΣΓ=1^)1/Ρ m, for all am+i,... ,an. If ρ < l,p ^ 0, (~22) holds. There is equality in both cases if and only if di = di, m 4-1 < i < n. D Assume that ρ > 1 and consider the left-hand side of (22) as function / of the variables am+i,..., an. Then /j = Ρζλ,, m 4-1 < J < n, where /■sr^n p\ — 1/p η η (Li=i ai°i) i=i i=i Since Ρ > 0 we have that the partial derivatives /' are zero if and only if the Qj are zero; that is if and only if —3— = ηΐ=1 \ , m + l<j<n. Equivalently, if "lb3 2^i=iaibi and only if —3— = ^ 1—7-, m 4-1 < j < n; that is a,- = a,·, m 4-1 < j < n. It is easily seen that this is a unique minimum of /. D Remark (i) As we have seen above, the case m = 1 is just (H). Remark (ii) A simple proof has been given in [Pecaric &; Beesack 1987a]. Another result of Beckenbach is the following. Theorem 17 Suppose ρ > 1, а > 0, β > 0, j > 0, and α, Ь are non-negative η-tuples; for m, 0 < m < n, define the (n — m)-tuple с by Ci = {otbi/β)ν /p, i = m-fl,...,n. Then β + ΐΈ^τη+,^ί β + ΊΣ^τη+ι^ί with equality if and only ifdi = Ci, m 4-1 < i < n. D η η β + Ί Σ aibi=a1'vtfa-1>v)+ Σ (71/ра0(71/гЧ) i=m+i i=m+i τι ■, ι η л ι ' <(α + 7 Σ αϊ) (α-ρ'/ρβρ'+Ί Σ bi) " > ЪУ (Н)' i=m+i i=ra+i i/p /? + 7LL+1te

198 Chapter III The case of equality follows from that of (H). D Remark (iii) A converse for this inequality has been given by Zhuang; [Zhuang 1993]. Remark (iv) All of these inequalities have been given considerable attention; [Iwamoto к Wang; Wang С L 1977,1979b,d,1981a,1982,1983,1988b]. 2.5.6 Ostrowski's Inequality This inequality is an extension of (C); [Ostrowski p.289]. Theorem 18 Let α and b be η tuples such that α φ b and deune the η-tuple с °У а. с = 0 and b.c= 1, then s-^n 2 n n n 4^<(Σ«?)(Σ^2)-(Σ^)2, with equality if and only if b + kY^aj-ak^aibi 1<fc<n. Remark (i) A proof can be found in [AI pp. 66-70], where several extensions due to Ky Fan & Todd are also proved; [Fan L· Todd]; see also [Madevski; Mitrovic 1973] Remark (ii) Beesack has pointed out that this result can be regarded as a special case of a Bessel inequality for non-orthonormal vectors; [Beesack 1975]] see also [Sikic L· Sikic]. It can also be regarded as an extension of 3.1.2 (7) in the case r = q = 2,s = 1. 2.5.7 Aczel-Lorentz inequalities As was pointed out in 2.4 (M) is essential for certain properties of norms on Rn. If a different norm is defined then we can ask if similar properties hold. A so-called Lorentz norm4 is defined as ΙΙ«ΐ£ = Κ-ΐ>ί)1/ρ,ρ>ΐ; this is defined on the set Lp of positive η-tuples for which α\ > (Σ™=2 α?)1/?; [ΒΒ pp.38-39]. Various inequalities using these expressions are then called Lorentz inequalities, [Wang С L 1984a]. However the first such inequality was proved by 4 This is Н.А.Lorentz; see [EM6 p.46-47].

Means and Their Inequalities 199 Aczel, [Aczel 1956b], so such inequalities have also been called Aczel inequalities] [DI pp. 16-17, PPT рр.Щ-126]. It turns out that the natural analogue in this situation is not (M) but (~M). As a result of this the Lorentz triangle inequalities are analogous to (^T), (~Tjv), and so the Lorentz norm is not a norm in the usual sense. The various Aczel-Lorentz inequalities are examples of reverse inequalities similar to those in I 4.4. THEOREM 19 (a) [Holder-Lorentz] If a,b are η-tuples in Lp, Lp> respectively, where ρ > 1 then aibi - X>fc > (a? -J2WPW' -Σ%')1/Ρ'- (23) i=i i=i i=i If 0 < ρ < 1 then (~23) holds; in either case there is equality if and only ifoP ~ bp . (b) [Minkowski-Lorentz] Ifa,b are η-tuples in Lp, ρ > 1, then ((«i + hY - £> + ЪУ)1/Р > (a? - ΣαΓ)1/ρ + (Ь? - Σ^)1/Ρ· (24) i=i i=i i=i If 0 < ρ < 1 then (~24) holds, and in either case there is equality if and only if а and b are dependent. (c) [Beckenbach-Lorentz] Suppose ρ > 1, а > 0, β > 0, j > Q, and α, b are non-negative η-tuples; for m, 0 < m < n, define the (n — m)-tuple с by Ci = {abi/β)ν lv, % = m 4-1,. .., n. Further assume that a — 7 Y^i=m+i af > 0, and that with equality if and only ifai = Ci,m + l < г < n. Remark (i) The case ρ = 2 of (23), the "Cauchy" case, is the original result of Aczel. Remark (ii) The above results are given by Wang С L but other authors have given different proof of certain results; [AI pp.57-59], [Mond &: Pecaric 1995b; Popoviciu 1959a; Wang С L 1984a]. 2.5.8 Various Results We first state an extension of (H) that is in [Mudholkar, Freimer Sz Subbaiah]. First note, as is shown in Lemma 1 of the quoted paper, that if b is a decreasing η-tuple, and m < n, then for some k, 0 < к < m — 1, 7ΓΓΊ - om_fc_!, Dm_fc < . (25;

200 Chapter III Theorem 20 If а and b are decreasing η-tuples, ρ > 1, m < n, then η m 1 / m—k — 1 D D 1 / / Σ«Α<(Σ«?) (Σ ЬГЧ(Б"-+Д-ГГ') ■ i=i i=i i=i ^ ' where к is given by (25). Remark (i) An extension of this result has been given in [Iwamoto, Tomkins &: Wang 1986b], and a simpler proof of this extension, using Steffensen's inequality, VI 1.3.6 Theorem 18, can be found in [Pearce Sz Pecaric 1995a]. The following result is in [Mikolas]; it reduces to (M) if pr = 1; see [AI p.369], [Alzer 199П]. Theorem 21 Ifaj, 1 < j < n, are m-tuples and 0 < r < 1, 0 < pr < 1, then m η η m ΣίΣ^'^^ΣίΣ^')· (26) If r > 1 and pr > 1 then (~26) holds. The next result is a problem set by Klamkin, [Klamkin &: Hashway]. It is a particular case of Callebaut's inequality, 2.5.4 Remark(iii). Theorem 22 If α and b are η-tuples and if η η f(k) = (Σα*2~4")(Σα^2~")' ° < fc < «; i=i i=i then f is decreasing. Remark (ii) The extreme inequality given by Theorem 22, /(0) > /(n), is just (C) The following is due to Abramovich; [Abramovich]. Theorem 23 If ρ > 1 and a,b,c are η-tuples with An = Cn = 1 and for some m, 1 < m < n, di > Ci, cij < Cj, -^ > -^-, 1 < г < m, m + 1 < j < n, then Oi Oj Σι/ριΐ/ρ' < V^Ti/У i/p where (d[i], · · · ^[n]) denotes a decreasing rearrangement of (d1}... dn). Remark (iii) On putting b = с this theorem reduces to 2.1 Theorem 4(b).

Means and Their Inequalities 201 Theorem 24 If a, b, u, y_ are η-tuples with и < v_ then η η η η η η {^ща!})* (Y^Uitfy -^ща{Ъ{ < {Υν^Ί)* (ΥνφϊΫ -ΥνΐαΑ. i=i i=i i=i i=i i=i i=i Remark (iv) This extension of (C), due to Dragomir & Arslanagic has been extended to (H) by Pecaric; [Dragomir L· Arslanagic 1991; Pecaric 1993]. The next result is in [Atanassov]. Theorem 25 Let α be an increasing η-tuple and k_ an increasing m-tuple of positive integers then TTi TL Ti г=± j = i j = i D The proof is by induction, using the inequality η AnBn < nYdjbi, where a, b are increasing real η-tuples. D Theorem 26 If α is an η-tuple such that a\ < 0,2/2 < ... < an/n and b is a decreasing η-tuple then г=1 г=1 i=i where αο = 0. There is equality if and only if αι = mi, 1 < i < n, and b is constant. Remark (v) This result of Alzer has a proof that is quite long and complicated; [Alzer 1992d, 1999b]. It generalizes the following variant of (C): η η η (]ГаЛ) <^Ьг^а^Ьг i=l i=l i=l The following is a very simple extension of (C) is due to Hovenier, [Hovenier 1994, 1995].

202 Chapter III Theorem 27 If 0 < ra < η — 1, and a,b are η-tuples deune ί ΣΓ=, «ibi + (EIUn+1 «г2)1/2(ЕГ=то+1 Ь?)1/2 if «г φ 0, 1(ΣΓ=1α?)1/2(ΣΓ=Λ2)1/2 ifm = 0. Then am < am-i, 0 < m < η — 1. D The proof uses the remark that if 1 < ra < η — 1 then αη_χ < am, because У^ <Ц^ = ^2 aibi + X] аг^ < Х^ aA + Π Remark (vi) The inequality αη_ι < ao is just (C). A similar extension of (H) has been made; [Abramovich, Mond Sz Pecaric 1995]. The next result is related to inequalities involving arithmetic means of powers rather than power means, quantities that have played an important role in statistics. 1 n Theorem 28 If α is a real η-tuple and am = - ^a™, m e N*, and if оц = 0, «2 = 1 then η i=l η — 1 1 а4>1+аз; <*2m > «m-i+°™; а3 < -?==, аА<п-2-\ -а. yjn — 1 η — 1 with equality in the last two inequalities if and only if a± = y/n — 1 and a^ = -l/v/n^T, 2 < к <n. The first result is in [Pearson], and given a different proof in [Chakrabarti; Wilkins]; these authors gave the bounds for the last two inequalities. The second inequality is in [Lakshmanamurti]; see also [Madevski]. Remark (vii) There are of course many other references of interest of which we mention the following: [MPF pp.83-189], [Abramovich L· Pecaric; Carroll, Cord- ner L· Evelyn; Dragomir 1987,1988,1994b; Dragomir &; Arslanagic 1992; Dragomir, Milosevic h Arslanagic; Dragomir L· Mond; Eliezer, Mond L· Pecaric; Liu; Mitri- novic L· Pecaric 1990b; Pales 1990c; Persson; Yang X]. 3 Inequalities Between the Power Means 3.1 The Power Mean Inequality

Means and Their Inequalities 203 3.1.1 The Basic Result The results of the previous section imply some relations between the power means introduced in the first section. Here we give the basic result, a fundamental generalization of (GA), called the power mean inequality. We will refer to this inequality as (r;s), where —oo < r < s < oo. Some authors refer to (r;s) as Jensen's inequality, see [HLP], [Beckenbach 1946; Cooper 1927a], although this name is usually given to (J). THEOREM 1 [Power Mean Inequality] Given two η-tuples a, w_ and r, s G R with r < s then алМ(&иО<яиМ(й;«г), (r;s) with equality if and only if α is constant. The first proof for arbitrary weights was given in 1879 by Besso, although the result had been formulated in 1840 by Bienayme; [Besso; Bienayme]. In the case of equal weights a proof was given in 1858 by Schlomilch for r and s positive integers or reciprocals of positive integers, [Schlomilch 1858b]. Later, 1888, a proof of the equal weight case for arbitrary r and s was given by Simon; [Simon]. In 1902 Pringsheim gave a proof for the case s = 1 and 0 < r < 1 that he attributed to Liiroth and Jensen; [Jensen 1906; Pringsheim]. D If r = — oo, or s = oo, (r;s) is just the strict internality of the power means, inequality 1(2). Using 1(4) we readily see that (GA) implies (0;s), 0 < s < oo, and (r;0), —oo < r < 0, and hence (r;s) when —oo <r<0<s<oo. Further a simple use of the second identity in 1(3) shows that (r;s), —oo < r < s < 0, follows from the case (r;s), 0 < r < s < oo . Another use of the identity 1 (4) shows that we need only consider the two cases (l;s), 1 < s < oo, and (r;l), 0 < r < 1. Finally suppose that we have proved (l;s), 1 < s < oo and that 0 < r < 1; then by (l;l/r) and 1(4) Ш[^/г]{аг;ш) > 21п(агш) = (Μ%\α;ν))τ, and by 1(4) Я^п (лГ;ш) — (^п(^;ж))Г- These two equations give (l;r), showing that we need only consider the case (l;s), 1 < s < oo. Following these remarks we give several proofs of (r;s). (г) Assume s > 1 and without loss in generality that Wn = 1. η Яп(й;«г) = ^(а|^)1/5Ч1_(1/8) < ®t\a;w), by (H), i=i proving (l;s).

204 Chapter III (ii) Assume without loss in generality that 1 < a^, 1 < г < n, Wn = 1, and that a is not constant and define Дг) = отИ(а;Ш), Ф(г) = r2^, r e R*. (1) Then *М = Ё(^)'».ь8(^)Г; И *'м = ,„,Γ Λί((Σ»'ί<)(Σ"'.°ί(1°8°<)')-(ΐ>«<ΐο8α,)!)<3) VZ-^=i ^^i J ^ г=1 г=1 г=1 ' Hence, by (С), sign0' = signr, which implies that ф(г) >0,г ^0. This in turn implies that f'(r) > 0, r Φ 0, from which (r;s) is immediate. This proof can be given without appealing to (C), [Burrows Sz Talbot; Wahlund]. (in) Make the same assumptions as in proof (г). Define the η-tuple b by b = ~—r. Then it is sufficient to prove that Vri°\b;w)>l. (4) From the definition of b, Σ™=1 Wibi = 1, so if we put Ьг = 1+Д, 1<г<п, then ft >-l and ΣΓ=ι ^<ft = 0- Hence, using (B), η η г=1 г=1 This immediately gives (4). See [Herman, Kucera & Simsa pp.167-168]. (iv)[Orts\ Assume that a is not constant. (i) First assume that r, s G N*. 9Я[!](а;ш) < Ш[„](а]ш) follows by (C). Now assume that m[£](a;w) < ЯЯ£+11(а; w), 1 < fc < m - 1. Then v2m (откт,(«;м)) m = (]>>αΓ) < (£>αΓ+1)(Σω<αΓ-1)> by (С), г=1 г=1 г=1 = (2rtLm+ll(«;»))m+1 (ш1[Г-1](а;ш))т_1 < I Ж[™+1] (а; щП ί Ш1Г] (а; щ)) , by the induction hypothesii This gives Wl^1 \a;w_) < Ш\™+1\а] w), and completes the proof for this case.

Means and Their Inequalities 205 (ii) Now suppose that r, s G Q, and that r = y/x, s = z/x where x,y,z G N* and у < ζ; then <(±m{a\iy)v,zr = т^,ш). i=l (iii) The general case of positive real r, s follows by taking limits, and as can easily be verified strict inequality is obtained. (υ) Assume that Wn = 1 and consider the extreme values of the function Y^=iWiCbi as a function of a subject to the condition Σ2=± wiai — A we easily see that if s > 1 then 2ln(&; w) < 9^n (й;ш)> while if s < 1 the opposite inequality holds. (vi) We now give an inductive proof of the equal weight case of (l;s) that has been attributed to Steinitz; see [Paasche]. In the case η = 2 assume that 0 < α < b when we have to show that 2((a + b)/2) < as + bs, or ((a + b)/2)s -as <bs- ((a + b)/2)s (5) This is immediate from the strict convexity of f{y) = ys, у > 0 and the property of I 4.1 Lemma 2. Alternatively, putting β = (b — a)/2, (5) can be written as {b-βγ-{b-2βγ <ь-{b-βγ which follows by noting that the function g(y) = ys — (y — /3)s, у > β, is strictly increasing, the last inequality being just g(b — β) < g(b). Now let α be a non-constant η-tuple, η > 3, and without loss in generality assume that max α = αη, mina = an_x when by the strict internality of the arithmetic mean an_x < 2ln(a) < an- Now write A = 2ln(a), an_x =A — ^, an = A + A when it is easily seen that (n - 1)A = Σ"_* <ц + (A - δ + Δ). Assuming (l;s) for all (n — l)-tuples this last identity gives П — 2 nAs = (n - 1)As + As < Σ at + (A - δ + Δ)' + A'· However (A - δ + A)s + As < (A - S)s + (A + A)s since this inequality can be written (A- δ + A)s - (A + As) < (Α- δ)3 - As which is a consequence either of the strict convexity of the function / above, or of the strict monotonicity of the function g above. This shows that nAs < Ση=1 af and completes the induction.

206 Chapter III (vii) Cauchy's proof of (GA), II 2.4.1 (zz), can be adapted to give a proof for the equal weight case of (r;s). Suppose that η = 2m+1, and we assume the result for η = 2 and η = 2m, then with a,b,c as in II 2.4.1 (zz), S^io) =^Ш^М < (^^+^^)1/S, by the case η = 2, <(№<»)' + №<*)'у'·, ^«„^ = Щ+1(а). The proof for all η is completed by the method of II 2.2.4 Lemma 6. (viii) The inequality to be proved is equivalent to Ып(а]Щ)) < 2tn(«s;^), s > 1, and so is an immediate consequence of (J), see II 1.1 Remark (xi), and the strict convexity of the function xs, s > 1. (ix) A simple proof can be given along the lines of Soloviov's proof of (GA), see II 2.4.5 proof (lix). By I 4.6 Example (vii) the function 7(a) = (Σ£=ι wiai) is strictly convex and homogeneous on (IR+)n. Now assume that Wn = 1 and then Vx(e) = w, so (l;s) follows from the support inequality, I 4.6 (24); [Soloviov]. (x) If, in the thermodynamic proof of (GA), II 2.4.5 proof (£), the heat capacity is allowed to be a function of the temperature then the argument there is readily generalized to give a proof of (r;s) in the case 0<r<s = r + l; see [Landsberg 1980b; Sidhu]. A similar idea occurs earlier in [Cashwell Sz Everett 1967,1968,1969]. (xi) Gao's proof of (GA), II 2.4.6 proof (Ixxii), can be adapted to give a proof of (r;s) in the case (l;s) and (r;l), r / 0; [Gao]. Assume without loss in generality that α is an increasing η-tuple with a\ ^ an and for ί G 1* let x_{ — i terms (ж,.. .,х,а»+1,...ап), and А (ж) = 21п(^;;ш) - ЯЯ^ж^ш), ,0<χ<αί+1, 1 < г < п. Then D[{x) = ^(l " (OTl"(f;m))1"t) < 0,by internality, 1(2). Hence A is strictly increasing if t < 1, and strictly decreasing if t > 1, and so, as in the quoted proof of (GA), Α (αϊ) > 0 if ί < 1, and Α (αϊ) < 0 if t > 1. (xii) A very simple proof of the equal weight case with r = 1 and s an integer can be found in [Brenner 1980].

Means and Their Inequalities 207 (xiii) A proof of the finite non-zero index case has been given by Qi; see VI 1.2.2 Remark(ix), [Qi, Mei, Xia &: Xu]. (xiv) A completely different proof is given by Segre, see VI 4.5 Example (iv); [Segre]. (xv) Another proof can be found in VI 4.6 Remark(vii); [Ben-Tal, Charnes L· Teboulle). (xvi) See also 4.1 Remark (iv). In all of these proofs the case of equality is easily discussed. D Remark (i) Proof (гг) is in [Norris 1935); an alternative calculus proof can be found in [Wang С L 1977]. A geometric version of the equal weight case of proof (viii) has been given in [Gagan]. An alternative proof of the case when r and s are integers can be found in [Ben-Chaim L· Rimer; Rimer L· Ben-Chaim]. Remark (ii) If0<r<s<oo and Wn < 1 then an easy deduction from (r;s) is that 7Z(r,a]w) < 7Z{s,a;w); further if Wn < 1 this inequality is strict; see 2.3 Remark (iii) and 2.3 Lemma 6(b). It is easy to see that if this inequality holds for all a then Wn < 1; see [HLP p.29). Remark (iii) If s = 2r then (r;s) is equivalent to (E«*<)2<Wn(f>af), which follows from (C); and the case of equality also is given by that of (C). Of course this is the basis of the first step in the induction in proof (iv) above. Remark (iv) Using s = 2r = l/2m, m G N, (r;s) gives yet another proof of (GA); [Schlomilch 1858a]. Assume α is not constant: then by (r;s) and 1 Theorem 2(b), М«;ш) = 9Як1](а;ш) > #41/2]te^) > · · ■ > lim #42_m](a) - 0п(а;щ). га—юо Remark (ν) If (GA) has been proved, without the case of equality Paley used an idea similar to that in the previous remark to complete the proof; see II 2.2.3 Lemma 5. Suppose then α is not constant, proceed as in Remark (iv): *п(а;ш) > Ж[п/2]&Ш) = (2in(a1/2;^))2 > (β№/2; "θ)* = ®п(аж), using the weaker form of (GA), obtained say by a limit argument from the rational weight case.

208 Chapter III Remark (vi) If r = 1, s = 2 the equal weight case of (r;s) is particularly easy to prove. η 2 n n (Σ°0 = Σai+2Σ °<oi ^ Σα?+Σ (a«2+ai)> ъу (GA)> i=i i=i i<i<j<n i=i i<i<j<n η =«Σαί· ΐ=ι See also [lies Sz Wilson]. Remark (vii) In II 2.2.1 Figure 3 the segment FD is Q(a, b), this is also the value of segment CD in II 2.2.1 Figure 4. Hence these diagrams give proofs of the η = 2 case of (r;s) for the values r, s = — 1, 0,1, 2. Remark (viii) If s = 1, r = 1/p < 1, w = cp ,aw_ = bp then (r;s) becomes n ^ ι /^ n 1 /*,' Σ^<(Σ»γ) (ЕЮ · i=l i=l i=l that is (H). Since several proofs of (r;s) are independent of (H) they give further proofs of (H). -Remark (ix) Ifl<r<s<oo then (r;s) is pictured geometrically in Figure 1, where 0 < αγ < · · · < αη, and the curve HL has equation χ = tr,y = ts, αχ <t<an, and R and S have coordinates 1/Wn Σ™=1 ща>1, 1/Wn Σ™=1 wia? respectively5, so G is the centroid of the points Pi...., Pn on the curve HL. 3 Q S s 1 4 x p / 1 V<-^-^ \y К G Η Ку У / |РП ] Jtv4) * Figure 1 See also VI 4.1 Example (iv)

Means and Their Inequalities 209 Remark (x) A proof of (r;s) in the case s = mr, m G N*, that uses Cebisev's inequality is given in 3.1.4 Remark (i). The inequality (r;s) with r = 1, s = 2 has been used to prove the following converse of (T). If π/4 < θ < π/2, a - θ < arg zk < a + Θ, 1 < к < η, then η 1 n V2; See [Janic, Keckic L· Vasic; Kocic L· Maksimovic; Vasic L· Keckic, 1971]. Burrows and Talbot have used proof (гг) of (r;s) to obtain another interesting inequality. If / is as defined in that proof then f'{r) > 0, and substituting in (2) bi = (аг//(0) > 1 ^ ^— n> leads to 1 n — ^Wibilogbi > 0. Wn . г=1 That is: Qin(b]w) = 1 =>► 2ln(blogb\ w_) > 0; or equivalently n / П \ l/Wn y£wibi = wn =* (ПГЬ<) >ΐ· г=1 г=1 There is equality if and only if b is constant; [Burrows L· Talbot]. The following result is due to Bronowski; [Bronowski]. If а > b > 1 and ifw. is a real η-tuple with Wn = 0 then Σΐ=1 aWi > ΣΓ=ι ^- Several proofs are given in the reference of which the following is the simplest. Let r > 1 be defined by a = br when: >(П(Ь№4)Ь") 1=1 , by (r-l;0), η l/\^n bWi > (J[ bWi^j i=1 = 1, since b > 1 and Wn = 0, г=1 Remark (xi) Inequality (r;s) can be used to give a very simple proof of 1 Theorem 2(b) in the case of equal weights; [Schaumberger 1992]. Note first that if r φ 0 then Ж[-г](а; аг) = Ж[£{а). So if r > 0, we have by (r;s) that <Зп(а;аг) > Ш^^д?) = Ш^(а) > 0η(α),

210 Chapter III and the inequalities are reversed if r < 0. Letting r —► 0 in these inequalities gives Umr_0OTW(o) = (5n(a). The function ra(r) = ( QJtJP (α; υ;) ] is strictly log-convex unless a is constant, when m is constant; see 2.3 Lemma 6(b), and 2.3 Remark(iii) The same result follows by considering φ of (1) since, using (3), ф'{г)/г = (logom(r)) ; see [Beckenbach 1966; Norris 1935]. The log-convexity property is due to Liapunov. Popoviciu has proved that log9JtJ^(a; υ;) is a convex function of 1/r, r > 0, and so 9Jt^ (a;w) is a convex function of 1/r, r > 0. This last result was pointed out by Beesack, who used it to give a simple proof of the following result due Hsu; see [Julia], [Beesack 1972; Hsu; Liapunov; Rahmail 1976]. Theorem 2 If0<r<s<t and α is not constant then г < ш[*Ча;ш)-Ш[:](а;ш) < s(t-r)m #4t] (a; w) ~ ®t] (а; w) r (* " s)' and if 0 < r < oo, t ^kr]fe^)-<~r]fe^) < ^n ЯИ^1 (а; Ш) - βη (a; w) minш' Remark (xii) The equal weight case of the second inequality was stated by Hsu and given a proof by Rahmail; the general case can be found in [Pecaric L· Wang]. This result should be compared with 6.4 Theorem 8. Remark (xiii) Further remarks and results on these convexity properties have been made in [Beckenbach 1942; Shniad]. Inequality (r;s), together with 1 Theorem 2(c), has been used to approximate roots of algebraic equations; [Netto pp.290-297], [Dunkel 1909/10]. Suppose for simplicity that all the roots are positive and denote them, in increasing order, by αι,...,αη. The numbers 9Я^(а), г е Z, form an increasing sequence with Hindoo 9Jt£J = an, lmv^.oo OJt^ = ab Now consider ( ( ) У^Кап°*2)г ) > r* Ε Ν*, we get an increasing sequence with 2 limit αη_ιαη; equivalently / \1/r ηΧ]!(α<1αί2)Γ

Means and Their Inequalities 211 In general / pEKuu^Y \1/r •-«»Ι(η-ρ+ΐ)Σί(Σ5=ι%) J The values of the numerator in (6) are easy to obtain if we take r = 2,4,8, This is done by forming equations whose roots are the sequences of those of the original equation, and iterating this process. If the fc-th equation by this procedure is xn + c\ x4*1 + ■ ■ ■ + Cn^x + ch = 0, and then the ratio in (6) is just (fc) \ V2* PCp (η-ρ+1)4*Λ In this way, by forming all such expressions, every root of the original equation can be approximated simultaneously. A different kind of generalization of Theorem 1 is the following result which contains the equal weight (r;s) as a special case; see [Marshall, Olkin Sz Proschan]. Theorem 3 If α and b are η-tuples such that b is decreasing, but b/α is increasing then ( Y™=1 a\ j Σ™=1 Щ) increases with r Various rewritings of (r;s) can be found in [Kirn Υ 2000b]. 3.1.2 Holder's Inequality Again In this section we will consider the inequality a«if](fib;м) < an|?](а;ш)яикг](кж). (7) Similar inequalities for power sums have been considered earlier; see 2.2 Remark (i), 2.3 Corollary 8. It is convenient to deal with various simple cases of this inequality and then state a theorem that will cover the remaining more interesting situations. As there is no loss in generality we will assume that q < r, q, r Ε Ш. Further since (7) is an identity if both α and b are constant we assume that this is not the case. (a) If either α or b is constant then (7) reduces to a case of (r;s). For instance, if α is constant, when by the above assumption b is not constant, then (7) holds as (s;q) if s < q, strictly if s < q, while (~7) holds if s > q, strictly if s > q. (b) If q = r = oo then (7) holds by I 2.2 (16) and (r;s), strictly unless s = oo and for some г, 1 < i < η, α^ = max a, bi = maxb. If q = r = —oo then (~7) holds by (r;s), strictly unless s = oo and for some г, 1 < i < η, αι = mina, bi = minb. (c) If q < r = oo then (7) holds strictly if s < q. If q = — oo < r then (~7) holds strictly if r < s.

212 Chapter III (d) If q = 0 then (7) holds for s = 0 and hence for all s < 0, strictly if s < 0 or if s = 0 and r > 0. (e) If r = 0 then (~7) holds for all s > 0, strictly if s > 0 or if s = 0 and g < 0. (f) If q = r = s = 0 then (7) is an identity. Theorem 4 If а, Ь and ш are η-tuples, ab not constant, then (7) holds if q, r,s £ R* and satisfy either (a) 0 < q < r and 1/q-{-1/r < l/s, or (b) q < 0 < r and 1/q + 1/V < l/s < 0. If qr < 0 < r and 1/q + 1/r > l/s > 0, or q < r < 0 and 1/g 4- 1/r > l/s then (~7) holds. The inequality is strict in both cases unless l/9 + l/r = l/s and a9-br. D Let - Η— = - then - Η— = 1 so t/q and t/r are conjugate indices. q r t q r In (a) £ > 0 and s < £ so ^](«Ь;ш) < Ш^(аЬ;Ш) =(21п((аЬ)';ш))1А, by(r;s) and 1 (4) <^i?]fe^)<r]&^), by(H) Similar arguments, using (H) or (~H), give the remaining cases. The cases of equality follow from that for (r;s) and (H). D Remark (i) The only case not obtained above is the case q = —r. Taking limits suggests that the correct result is minab < Wl[~r\a; w)Wt$(b\w) < maxab. By the use of 2.1 Corollary 2 inequality (7) can be generalized. Corollary 5 If η > 0, 1 < i < m, l/s > Y™=11/n and if a^\ 1 < г < m,w, are η-tuples with Π2=ι -^ no^ cons^an^ then m m <s](nfl(i)^)<nKril(fi(i)^)), with equality if and only if l/s = Y^=1 1/г{ an<^ {^) \ 1 < i < m are proportional. Remark (ii) See also IV 2 Theorem 12 for another way of stating this result. It has been remarked earlier that certain of the basic inequalities are equivalent. It is convenient to collect these equivalencies here.

Means and Their Inequalities 213 Theorem 6 (a) (B) 4=^ (GA): (b) (GA)=* (H): (c) (H) <=> (C): (d) (H) <*=► (r;s): (e) (H) =» (M): (f) (M) =» (T); (g) (T)=» (C); (h) (r;s), r, s > 0, =» (GA). D (a) For the one implication see II 2.4.5 proof {lvi)\ for the other implication let 0 < a < 1, then using (GA) we have: (1 + x)a = (1 + ж)а11_а < α(1 + x) + (1 - α)1 = 1 + аж. (b) 2.1 Theorem 1, proof (г). (с) 2.2 Theorem 5. (d) 3.1.1 Theorem 1, proof (г) and 3.1.1 Remark (viii). (e) 2.4 Theorem 9 proof (г). (f ) See 2.4 (g)(TN) with p = 2 is N X>i + bi) 2 < Σ«? + N Έ*ι which on squaring gives (C). (h) Take s = 1 in (r;s) and let r —> 0. D Remark (iii) Better results are possible since for instance (GA) is equivalent to (GA) with equal weights; (B) with 0 < a < 1 is equivalent to (B) with a > 1; etc. This topic is discussed in [MPF pp. 191-209], [Infantozzi; Maligranda 2001]. 3.1.3 Minkowski's Inequality Again Theorem 7 (a) Let a, b and ш be η-tuples and suppose that 1 < r < oo, then OTlrl (a + Ь; «г) < QJtM (a;Ж) + JWjT1 fe «г); (8) witJh equality if and only if either (i) r = 1, or (iij 1 < r < oo and α ~ b, or (iii) r = oo and for some г, 1 < г < η, max α = α^ and maxb = bi. If r < 1 then (~8) holds. (b) If a^ = (ali}..., amj) > 0, 1 < j < n, and a,^ = (a^,..., a^n) > 0, 1 < г < га, и an m-tuple and v_ an η-tuple, and suppose r,s Gl, r < s, then, *(^](^;.);^) < m^(m^\a{iy,v);u). (9) Π (a) If r φ 0 then this is just (M), 2.4 Theorem 9 and Remark (i). Alternatively we can use I 4.6 Example (vii) which shows that M(a) = 9Л^(а;ш) : (R* )n i-> Ш is strictly convex. The case r = 0 follows by taking limits.

214 Chapter III However these methods fail to give the cases of equality when r = 0 so we give two independent proofs of this case. (г) By (GA) in a quasi-linearization form, see 2.1 Remark (vii), and assuming without loss in generality that Wn = 1, ®n(a + k\w) = inf y^Wid(ai + bi), where С = {с; 0n(c; w) = 1}. cec z—' But infcec Σ^=ι wici(ai + Μ > infcec Σ)Γ=ι ^c*a* + infcec ZlLi ^c^, by I 2.2 (e)(~15) so we get ®n(a + k]w) >0n(a;w) + 0n(&;w)- (10) the case r = 0 of (a). More simply: with с = α or Ь and using (GA), в"(^ш) ^6п(с/(а + Ь);ш)<^п(с/(а + Ь);^); ©η (α 4- о; w) now add the two inequalities obtained by taking the two values of c. The case of equality follows from that of (GA). (гг) Since by I 4.6 Example (viii) the function χ(α) = 0п(&;ж) is strictly concave (10) follows from II 4.6 (-19); see [Soloviov]. (b) Assume r, s G t, rs / 0, r < s, and without loss in generality that Um = Vn = 1 when (9) is η m J mm , j = i i=i i=i i=i or η m ι πι η (E(E«^;4)s/r) <Σ(Σ«;/ρ^)ρΛ; but this is just 2.4(16). The other cases are proved by taking limits. D Remark (i) The cases of equality in (9) are discussed in [HLP p.31]. This inequality is sometimes called Jessen's inequality, [MPF p. 108]; it has been much generalized, see [Toader 1987b; Toader L· Dragomir]. Remark (ii) These results have been proved by many authors; see [BB p.26], [Beckenbach 1942; Besso; Bienayme; Giaccardi 1955; Jessen 1931a; Liapunov; Norris 1935; Schlomilch 1858b; Simon]. Remark (iii) Inequality (10) is sometimes called Holder's inequality. A simple proof of (10) can be given using the inverse geometric means of Nanjun- diah, II 3.4. The proof in fact gives a Rado-Popoviciu type extension of (10).

Means and Their Inequalities 215 Theorem 8 If η > 1 then (®n(a;w) + <3n(b;w)\Wn < /gn-i(a;w) + gn-i(fcw) with equality only if αη0η_ι(6; w) = bn0n_i(a;w;) D By II 3.4 (22) and II 3.4 Lemma 10 (a) which gives the above inequality. The case of equality follows from that of II 3.4 Lemma 10 (c). D Remark (iv) A simple proof of the right-hand inequality of II 2.5.3(36) can be given using (10). 1 + 6п(а;ш) =®n(e;w) + <6n(a;w) < 6n(e-f-а;;ш), by (10), <2in(e + a;w;), by (GA), = 1 + 2ln(a;w). More can be proved by using (8) rather than (10), and appealing to (r;s) instead of (GA). If 0<r < 1 then 1 +Ж[£]{а;ш) < ЯЯ|Г](е + а;ш) < 1 + Яп(а;ш). Conversely, using the equal weight case of left-hand inequality of II 2.5.3(36) we can prove the equal weight case of (10); [Dragomir, Comanescu L· Pearce]. ^ = 1 + в„(а/й < 6n(e + а/Ь) = ^j^- 3.1.4 Cebisev's Inequality We can extend Cebisev's inequality, II 5.3 (1), to power means. Theorem 9 (a) If 0 < r < oo, and if a, b and w. are positive η-tuples with α and b similarly ordered then, m^(a;w)m^(b;w) < m^(ab;w), (11) If a and b are oppositely ordered then (~11) holds. (b) If —oo < r < 0, and if α and b oppositely [similarly], ordered, then (11) , УУ-П-

216 Chapter III [(-11)}, holds. There is equality in (11) when r = 0, and in all other cases there is equality if and only if either a, or b is constant. D The case r = 0 is trivial and is in any case pointed out in II 1.2 (9). If r ^ 0 then by 1(4) we need only consider the case r — 1. This however is just II 5.3 Theorem 4. D Remark (i) If the η-tuples a^, 1 < к < m are similarly ordered and r > 0 then from (11) m m fc=l fc=l Now putting s = mr and a^ = · · · = oSm^ this inequality is just (r;s). 3.2 Refinements of the Power Mean Inequality The inequality (r;s) has a similar form to (GA) and so it is natural to consider extensions similar to those considered in II 3. 3.2.1 The Power Mean Inequality with General Weights As we have seen, when r, s Ε Ш the inequality (r;s) is a particular case of (J), so we can use the Jensen-Steffensen inequality, I 4.3, to extend II 3.7 Theorem 22 to allow general weights in (r;s) that satisfy II 3.7 (49). In this more general situation however we must take full note of the remarks in I 4.2 theorem 12 and I 4.3 Theorem 19 and ensure that we are in the domains of the functions being used; that is we must require that the sums Σ™=1 Wia\, Σ™=1 ^α| be positive. We will not pursue this further but consider the case η = 2, that is the extension of II 3.7 Theorem 23. Theorem 10 Ifn = 2, u>iu>2 < 0, W2 = 1 α a 2-tuple, r, s e M*, r < s, and if wia[ -f ^2^2 > 0, wia{ 4- ^2^2 > 0 then (~r;s) holds. Remark (i) If we assume that a\ < a2 this theorem can be put more simply since then all we need is that wia{ 4- ^2^2 > 0, or w2 > ~αί/(α2 — αι)· Further we can extend II 3.7 (51) and (52) as follows: if w2 < 0 then Wl[s]{a; w) < Ш[г]{а\ w) < au while if ил < 0 then m[r](a;w) > Wl[s\a;w) > a2. Remark (ii) This simple theorem is behind the interesting inequalities of Nan- jundiah, see below 3.2.6. 3.2.2 Different Weight Extension

Means and Their Inequalities 217 Theorem 11 If а, и and v_ are η-tuples, and ifr,s e R*, r < s, define the η-tuple w_ by 2us~r m\r]{a.u)<^^mls]{a;vl with equality if and only if aw_ is constant. D Since m[£\a;u) _ fm^(aw-us/^-^vr/{r-s))\ (fWfc](w;u) ΐΣ^Ι^ίΣ^^'^ί^^' the result is an immediate consequence of (r;s). D Remark (i) See [Bullen 1967; Mitrinovic &: Vasic 1966a]. The following theorem is an application of this result;[Vasic L· Keckic 1971]. THEOREM 12 If z_is a complex η-tuple, w_ a positive η-tuple, and ifp> 1 then fc=l fc=l with equality if and only ifw. \z\p~x is constant, and z-^zj > 0, 1 < j, к < п. A direct proof of this inequality using (H) can be found in [Bullen 1972]. 3.2.3 Extensions of Rado-Popoviciu Type An extension of (R) and (P) to power means is given by the following result. Theorem 13 Assume that α and w. &re η-tuples, η > 2, and r,s Ε R, r =£ s. (a) Ifr<s and λ = r or s but A^O, then ,λ / Μ Ν λ^ ИЦ(«ЩМ(а;«г)) -(Ш4г](а;«г))' with equality when λ = s if and only if ап = Ш^_г(а;г^1), &nd when λ = r if and only if an = 9Jt£! 1 (a; w). (b) Ifr<Q<s then f№teuu\W\ (nW-temiy*-1 f (13) аякг)(а;ш)У WtLtem)

218 Chapter III with equality, when s = 0, if and only if an = ^tjjl-^aju;), when r = Q, if and only if ап = Шп_г(дг,т), and when r < 0 < s, if and only if both conditions hold. D (a) The case r ^ 0, s = oo, when λ = r, or s ^ 0, r = — oo, when λ = s are almost immediate. As in 3.1 Theorem 1 we can if s =£ 0 use 1(4) to assume s — 1, or if r =fi 0 that r = 1. Further if either s = 0, or r = 0, then (12) reduces to (R). So it suffices to consider the cases (i) r = 1 < s < oo, and (ii) —oo <r<l = s,r^0. Case (i) In this case if λ = s then (12) is equivalent to η —ι η W^^uW) +»rKan)S>Cs(Ew*e·) - i=i i=i and when λ = 1 to ^-i/s(i>«01/s > nc;/s(x><f/s+<-i/sko1/s. i=i г—ι An application of (B), or (H), confirms these inequalities and gives the cases of equality. Case (ii) To consider this case put χ = αη assume that λ = s = 1, and put f(x) = Stn fe Ж) - ^1г](а;ш)· Then «*>- (If *·-<*»+£*) - (^(*.<-)Г+^)"Г. and ^"и^Д1 Wn+ wn[ χ > ) )■ So / has a unique minimum at χ = x0 = 9Jt£L x (а; ш) · As a result f(x) > f(xo) if χ φ жо> and simple calculations show that this is the desired result. (b) The cases of non-finite r and, or, s are straightforward. Further, if r, s e Ш and either r or s is zero the result reduces to (P) by using 1(4). So we may assume

Means and Their Inequalities 219 that r,s el*. Then , /9^femhw" w/l, (Wn_4 1 ^ wn sn log(^R(^)) =4slogbrfc|>^+^g by the concavity of the logarithmic function, The case of equality follows since the logarithmic function is strictly concave. D Remark (i) Another proof of one case of part (a) of this theorem is given below, see 3.2.6 Theorem 24. Remark (ii) The technique used in II 4.1 Theorem 1 proof (г) can be applied to the proof of (a) case (ii) to obtain inequalities converse to (r;s). This will not be done here as the whole topic will be taken up in 4 below. Remark (Hi) Many similar results follow from more general theorems to be proved later, see IV 3.2, and so we will not discuss them at this point; see [Bullen 1968; Mitrinovic L· Vasic 1966a,b,c], A simple deduction from Theorem 13 is the following. Corollary 14 If α and гц are η-tuples, n>2, and —oo <r<l<s<oo, r ^ s then Wn(<mW(a;w) - Tl^(a-w)) > Wn^(Tl[:Ua;w) - fB^l^w)), (14) with equality when s = 1 if and only if αη = 9#£_ χ (α; w), when r = 1 if and only if an = 3Jt£Li(a;w)j and when r < 1 < s if and only if both of these conditions hold. D Take λ = r = 1 and A = s = lin(12) and add the resulting inequalities. The cases of equality are immediate. D Remark (iv) Inequality (14) and its analogue (13) are the simplest extensions of (R) and (P).

220 Chapter III If we do not assume that r < 1 < s then (14) need not hold as the following example of Diananda shows. Example (i) Assume that 1 < r < s < oo, 0 < J < 1, wi = · · · = wn = 1, η > 3, ax = · · · = an_3 = an = 1, an_2 = (1-f ^)1/s, an_x = (1 — £)1//δ. Then (14) reduces to 1 + (n - 1)^1 x (a; w) > nQJlMfeu;) or equivalent^ 2ln(b) > mt|T](b), where the η-tuple Ь = (fc,..., fc, 1), with к = (η - Ι)1/7, (η - 3 4- (1 4- £)r/s + (1 - £)r/s). Since fc =£ 1 this is false by (r;s). If r < s < 1 repeat the above replacing s by r in an_2, and in αη_ι. 3.2.4 Index Set Extensions We now prove a general theorem that implies many extensions of the results in II 3.2.2 ; see [Mitrinovic L· Vasic 1968d]. Theorem 15 Let λ, μ > 0, λ 4- μ > 1, α, Ь, η, ν sequences, /ι, /2, Λ, «^2 index sets with Ιι, Ji disjoint, I2, J2 disjoint, then ^ил*£ша (<1Л (й; М)) λΓ«U (6; «))"' (15) If λ 4- μ > 1 (^15^) is strict; if λ + μ = 1 (^15^) is strict unless и№№!(я4)*№£ыГ = ^^K|fe«))JrKi(^))^. (16) If Χμ < 0, λ 4- μ = 1 then (~15) holds, with the same conditions for equality. D This follows immediately from 2.3 Corollary 8(b) with η = 2 on putting Λ = 1/p, μ = 1/У and a\ = Uh (9Л^(а; u))r, αξ = г/л(ОТ^(а;гг))г; and Ь?' = ^/2 (^й (& 2>))S > b2' = V>2 (9Я^ (6; v))S. The cases of equality are immediate. D Remark (i) Since Theorem 15 is such an immediate consequence of a very simple case of (H) any deduction from this theorem can also be obtained directly and easily from (H). Remark (ii) Inequality (15) is easily extended to allow for m pairs of disjoint index sets. Remark (Hi) Put \λ = I2 = Ι, 3γ = J2 = J, r = p, s = ρ', λ = 1/p, μ = 1/p' y щ = 1 = Vi, г G N* then (15) becomes (Σ«?Γ( Σ »Г)"^(Е°г)"'(Е'Г)"^(Е<.?)"'(ЕьГ)"''. Since clearly У^ CLibi = ^2 aibi + X] аЛ /UJ / J the last inequality implies the first part of 2.5.2 Theorem 12, and the case of equality in that theorem is an easy consequence of that in 2.4 Corollary 10.

Means and Their Inequalities 221 Corollary 16 If I, J are disjoint index sets, a, u, v_ are sequences, r,s Gl, r < 0 < s, then - Vr/(s-r) у m[s] fe ^ J yr/(s-r) у m[s] fe ^ The inequality is strict unless Ui(m[f](g;u))r _Uj(m[?(g;u))r Vi (Μψ(α;ν)Υ VJ (Mlf (α; v))s ' If rs > 0, r ^ s (~17) holds and is strict under the same conditions. D This is an immediate consequence of Theorem 15, by taking α = 6, I\ = h = I, Ji = J2 = J, λ = s/(s - r), μ = -r/(s - r). D Remark (iv) Defining the following function on the index sets 4/)=er)iKr|yrVS/i!"r) vr/(s-r) \(mW(a;v))s then Corollary 16 says that ν is super-additive if rs < 0 and sub-additive if rs > 0. Remark (v) Taking I = {1,2,..., η — 1}, J = {n} (17) becomes with equality if and only if (18) \unVn-Jan (OT^feu))"' Remark (vi) Inequalities of this type for power means seem to have been first discussed by McLaughlin & Metcalf, and independently by Bullen; see [Bullen 1968; McLaughlin L· Metcalf 1967a,b,c].

222 Chapter III Corollary 17 If I, J,a,u,n are as in Corollary 16 and r < 0 < s then ι] Ъ Uiuj (Wi{?Oj{a;u)y\U'UJ fUi^^UJyV (Vj(W№(&1^ ' ^ Vruj V jrnW J(fi; v) > ) -\VA m[f (a; v) > ) γλ Ж1? (a; v) (19) Inequality (19) is strict unless %Rj(a]u) = %Rj (a;u). D The right-hand side of (17), divided by Uiuj can be written in the form Ui Ui (Ш1Р(а;и)у\ | U j lUj/mp{a;u Utuj \ Vi \ mlf (a; v) > ) Uiv J \ Vj \ mlf (fi; „ which by (GA) is greater than or equal to rUi/(s-r)Uiuj / <,r\, ч \ rUj/{s-r)Uiuj Uj (Tl[P(a;u)y\ Uj (Щ*(а;и)\а Ут\<тЫ(п-,л) / \Vr\ Vi Km[jS]{a',v)J J \VJ KWl[j\a;v)< On taking suitable powers of this inequality (19) follows. D Remark (vii) Taking J = {1,..., n}, J = {n 4- 1, ■ · ·, η + га}, η = ν, s = 1 and then letting r —> 0—, (19) reduces to II 3.2.2 Theorem 8(b), together with the case of equality, making Corollary 17 a generalization of that result. Define the following function on the index sets, σ(Ι) = Wif№j(a;w), then the following is an analogue of 2.5.2 Theorem 12; [Everitt 1963). Corollary 18 If s > 1 and I, J are disjoint index sets then σ(/ϋ J) >σ(/)+σ(7), (20) with equality if and only if Τϊψ (or, w) = Tl[J](a;w). If s < 1 then (~ 20) holds with the same case of equality If s = 1 then (20) is an identity D The case s = 1 is trivial so suppose first that s > 1. In Theorem 15 take Д = Д = J, Ji = J2 = </, r = 0, λ = 1 — μ, μ = 1/s, and и = у_ = иг, the result is then immediate. If s < 1, s =£ 0 there is a similar proof. If s = 0 the right-hand side of (20) is \ vviuj W/u j J <WiOJidi{gr^Wjlw^ej{a:^ , by (GA), =WiuJ<6IuJ(a-w). The case of equality follows from that of (GA). D The following generalizes II 3.2.2 Theorem 8 and the notation used is introduced there; see [Bullen 1968; McLaughlin L· Metcalf 1967b, 1968b; Mitrinovic L· Vasic, 1968d).

Means and Their Inequalities 223 Theorem 19 If a,u and ν are (n + m)-tuples, A e 1, 0 < Xr^s-^Um < Un+1} s/r > 1 then Уп+тЫ:[т^у)Г -^шт{й.^у (21) U m '^(^(α;*))' -Щ^Ы;\а;и)У(ип+т-итХ^-^а)/г + (aJt1M)'4(5t1M)ii and ^KU^))5 -A^Kfe^tf^)8 (22) If 0 < s/r < 1 then (~ 21) and (~22) hold. Equality occurs if s = Q, when no restriction need be placed on X, or if s =£ 0 if иМ;](а;и) = (Хг/(г-8)ип^ш-ит)1/гт^(а;и). Letting r —» 0 in (21), (21), gives inequalities that hold for all s, that are strict except under the conditions obtained from the above conditions on letting r —» 0. D Proof of (21) case (i) :r φ 0, s =£ 0. In this case (21) is equivalent to 1 n+m jj n+m 1 V^ s Un+m / V~^ r\s/r Vm i=i Um i=i n+ra 1 n+m V ^ \TTS This, in turn, is equivalent to + = V VioJ - -±r( У щагЛ'/г. Vm /-^ \TtsIt ^ ' m *=n+i MJm i=n+i (У^ща1У,г(ип+т -um\r^f-s),r (23) n-\-m n+m ι=η+ι ΐ=ι and this last inequality follows from the η = 2 case of (H), as does the case of equality.

224 Chapter III Proof of (21) case (ii):r^0, s = 0. Now (21) is equivalent to Y^n_u^n^K_i{Un+m_UmX-l) + 1 + x-li which reduces to an identity for all λ. Equivalently we could use (23). Proof of (21) case (iii) :r = 0, s φ 0. Letting r —► 0 (21) becomes %^KSU«;^))5 -4^Un+m{a-u)Y (24) V m U m V m U m ' + (Ш[^(а;у)У -A-^e^feu))*· But (24) is equivalent to 1 n-\-m jj ч л n TT J_^Via?-%t-(en+m(a;«)) >^-^^-г(в«М) λ""/"» 1 n+m s _ + ^~ Σ ^-^(«nfeu)) λ^/υ", (25) or to which is a special case of (GA). The case of equality follows from that of (GA). Proof of (20) case (iv) r = 0, s = 0. This case is covered by the previous case where the assumption s =fi 0 is not used. Proof of (22) In all four cases (22) holds under the same circumstances as (21); we see this as follows in two important cases. Case r^0,s^0. After some simplification (22) is seen to be equivalent to (21). Case r = 0, s =fi 0. Letting r —> 0 (22) becomes: K+mUl ,„.„\Y yUn+mf,* f„...\Y (WZ(&y) \m^m{a,v)) -A^((3n+m(a;n)) Vrn V J Um V / у <Dm{a]V > ^.(«шк-'teiO)' -л—^^(en(iEil))J (ψτ^ which after some simplification is seen to be equivalent to (25). D

Means and Their Inequalities 225 Remark (viii) It should be remarked that neither (21) nor (22) depend on v_. Remark (ix) If we take λ = 1, u = ν then (22) shows that if s/r > 1 then σ(Ι) = [/.((OTWfeu))8 - {m[p(a;u))s) is super-additive, which generalizes II 3.2.2 Theorem 7. Included as special cases of these results are the inequalities in II 3.2.1, in particular those of Example (iii); and for further results see below 3.2.5 and IV 3.2. 3.2.5 The Limit Theorem of Everitt The result of II 3.3 is easily extended to power means as was pointed out by [Everitt 1967,1969]. His results are included in the following theorem, [Diananda 1973] Theorem 20 If α is sequence and r, s are real numbers with —oo < r < 1 < s < oo, r =£ s, then Итп_»оо η(9Jt|^(a) — 9JtJP(a)) is unite if and only if: either (a) r = 1 and Σ™=1 αη converges , or (b) for some positive α, ]C^ULi(an — ol)2 converges. The above limit exists because of 3.2.3 (14), but as we saw in 3.2.3 Example (i) the limit need not exist if r and s do not straddle 1. However it could still happen that Theorem 20 holds under these circumstances. A partial answer to this question was given by Diananda. THEOREM 21 If —oo < r < s < oo and the sequence α converges to the positive number а then limn^oo n(Wl[£] (α) - Τ$ (a)) exists and is finite if and only if the series Υ^=ι{αη — οι)2 converges. In the same paper Diananda discusses the analogous Нтгг_>00(ЗЯ^(а)/ЗЯ^(а))п. By 3.2.3 (13) this limit exists if —oo <r<0<s<oo, r^s, but not in general if r and s do not straddle 0. Theorem 22 If α is sequence and r, s satisfy —oo<r<0<s<oo,r^ s, then \1тп^^(т^](а)/тР(а))п is unite if and only if for some positive a, ^=1(an — a)2 converges. If lrnin^oo an = a, 0 < a < oo, the limit exists, finite, under the same condition if —oo < r < s < oo. 3.2.6 Nanjundiah Inequalities As might be expected the results of II 3.4 can be extended to power means. The arguments being similar to the earlier ones will not always be given in detail.

226 Chapter III If r G R, then the Nanjundiah inverse rth power mean, or just rth Nanjundiah mean, of the sequence α with weight ш is: U"1^), ifr = 0. Of course 9l|^ (α; w) ~ 2l~γ (α; υ;), and we define the sequences of rth power means Шгча; w), and inverse power means 21 (а;ш)> in a manner analogous to that used in II 3.4 for the cases r = 0,1. We easily extend II 3.4 Lemma 10 (a) and (b). Lemma 23 (a) With the above notations m[r](vx[r];w) = m[r](m[r]-w) = αη, η e n*. (b) If η > 1 and r,sGl with r < s, and ifWnasn, η £ N is increasing, then Steffi) >*М (а;ш), (26) with equality if and only ifa^l_1 = asn. D (a) Elementary. (b) First note that 9lJ^(a;u;) = ^r\an-1,an\—Wn-1/wn,Wn/wn), and apply 3.2.1 Theorem 10, noting 3.2.1 Remark (i). D We can now give a very simple extension of (R), a particular case of 3.2.3 Theorem 13 (a). Theorem 24 Ifr,seR with r < s then , ч (27) > w^m^w))8 - Kite™))5), with equality if and only if an = 9Jt£|_i(a; ш)- D This follows as in proof (v) of II 3.1 Theorem 1, but using (26) and noting that Wn(Wl[*]y, η G N, is increasing; <sl Ш:]; ж) = an = этм (щгм; ш) > я!>] Ш:]; ш), which is what we have to prove. The case of equality follows from that for Lemma 23(b). D We can also give an analogue for (M), and use it to prove a Rado type extension of (M).

Means and Their Inequalities 227 Theorem 25 If r > 1 and Wnan, Wnbn, η e N are increasing then <r] (a + Ь; ш) > <rl (a; ш) + <r] (& щ); (28) if r < 1 then (~28J holds. D Assume that r > 1 and let ttn , dn = \ , C^n = Wn{an + 6n)r,nn = AUn, η e N*. on + bn an + 6n Simple calculations then give: т^шн^шш)=<r]fe30+<r](j;a)<8t-i(sg)+a-1(d;a)> by(26) = а-1(с + 4;гг) = 1. D Corollary 26 If r > 1 then for η > 1, ^п((аг4г](а;ш) +^lr]fe^))r - (5Я1г](а + 6;ш))Г) (29) D Replace а, Ь by 22м (а; ш), 22[r] (а; ш) respectively in (28). D We now will generalize II 3.4 Lemma 11 and Corollary 13. Lemma 27 If r, s e R, r < s then *M (2I[s]; ш) > ^ln] (ffiW; Ш)■ (30) D We can assume that r,sGl* since the other cases follow from II 3.4 Lemma 11. Further we will assume that r = 1, s > 1, when (30) is Wn/Wn s _Wn-1 s Y^s _Wn~1fWn~1 s _Wn-1 s \ Ι &η Qjrr-, — Λ Ι Ι (λη — Λ &П — 2 I Wn \Wn Wn J Wn νΐϋη_! Wn-i J ^(wn(wn wn-i γ Wn-./Wn-, wn-, y\1/s \wn \wn wn / wn ^гУп-i гУп-ι ' / or putting, an = a, an-i = b, an_2 = с and ρ = Wn/wn, q = Wn-i/wn-i, p(pas - (p - 1)6*)1/S - (p - 1) fab' -(q- l)cs)1/s / \ l/s (31) >(p(pa - (p - 1)6)* - (P - 1)(?6 - (9 - l)c)S) ·

228 Chapter III Now if (p - 1)7 = (p — q)b + (q — l)c the right-hand side of (31) can be written ((p(po - (p - 1)6)s - (p - 1) {pb - (p - 1)7)S)1/S, which by (28) is less than or equal to p(pa° - (p - l)fos)1/s - (p - l)(p6s - (p - 1)7S)VS. So we will have proved (31) if we show that pbs-(p-l)7s>4bs-(<Z-l)cs; equivalent ly Vp — 1 ρ — 1 / ρ — 1 ρ—1 This however follows from (J) and the convexity of f(x) = xs. D Theorem 28 Ifr,seR,r<s and η > 1 then wn{m^\m^;w) -яяМ(Ш£М;ш)) > Wn^m^m^-w) -artLeLi@i[rl;M)). D The proof follows that of II 3.4 Theorem 15, but using Lemma 27, rather than II 3.4 Lemma 11. D Corollary 29 If r, s e R, r < s and η > 1 then msn№[r]; ш) < ®t] (ti[s] -ш), r<s. (32) D Immediate consequence of Theorem 28. D Remark (iii) The above results are stated in [Nanjundiah 1952] with some extra conditions on the weights, and was proved in the author's unpublished thesis that was communicated privately to the author who then published Nanjundiah's proof; [Bullen 1996b]. A different proof given in [Rassias pp.27-37], [Mond L· Pecaric 1996a,c; Tarnavas Sz Tarnavas] is based on the lemma of Kedlaya, VI 5 Lemma 5(a); [Kedlaya 1994, 1999]. The following deduction from (32) was made by Nanjundiah. COROLLARY 30 [Hardy's Inequality] If ρ > 1 and b is an η-tuple then яикр1(Ш)<-^алМ®, or J2^(k)<(-P-YJ2^-

Means and Their Inequalities 229 D In the equal weight case of (32) take s = 1, r = 1/p and a = Ψ when we get : where the notation Si(p, b) is defined in 2.3. Now if we put w = (l, 2-1/p,.. . n~1//p), and <S(p, b) = (Si(p,k), 1 < г < η), this becomes i=l Σ Я"® < n^W* Ш^ (S(p, b);w), , <v}-pWPSn{p,b), by internality, 1(2), =i(i>-i/p)pi>? ПР~ · 1 г—1 г—1 <(~ΓΣ>Ρ· D Remark (iv) This inequality has been the object of considerable research; for further information see [AI p.131; DI pp.111-112; EM4 p.369; HLP pp.229-239]. 4 Converse Inequalities We extend the discussion of II 4 by considering the quantities defined for all r, s £ Ш. Then the power mean inequality, (r;s), says that if r < s then Q;'s(a;^)>l; ^s(a;w)>Q. with equality if and only if either r = s or a is constant. We are looking for upper bounds for these quantities assuming that 0<ra<a<M<oo; (1) upper bounds that depend only on Μ and m and that improve the trivial ones ^ > Q£s(«; Ш); М-т> Вг^(а;ш).

230 Chapter III Various other forms of converse inequalities are possible; consider for instance 3.1.1 Figure 1. We have the simple inequalities RH < RG < RK and SI < SG < SL. These imply that if 1 < r < s < со, a is not a constant and (1) holds, then ((%г^) <**w+MS™;-MrmS)1/r < *№) \\MS -ms J K n w/ Ms-ms J n w <™Ы / ч { (Ms -ms\ / frl, N\ * Mrms - Msmr \ 1/s Such inequalities have been used, for instance in actuarial mathematics; see [Black- well & Gir'Schick p.31], [Giaccardi 1955]. 4.1 Ratios of Power Means Theorem 1 (a) If r, s e R*, r < s then 1<г<п (b) If r, s в R*, r < s and for some fc, 1 < к < η, Tt^\a;w) < ak < SOT^fe w), then Q%s(a;w)< ши^Йй). l<«<n D (a) it is easily seen that l/s (n -1)-1 Y?i=1(wn - «*)(ш£!_1&;«0)г) (" -1)"1 ΣΓ=1(^η - ^(ялЦ^-.ш)) ((η - i)-1^.^ " «Ό«1.(βί;«;))β) S > mm Q^fe), 1<г<п by (r;s), and by noting that (n — Ι)-1 Σ™=1(^η — ^) = 1· (b) Assume, without loss in generality, that a is increasing, when the hypothesis implies that η Wn{Tl^(a;w))s < £><α? + wk(Tl^(a;w))s, г=1

Means and Their Inequalities 231 and so Similarly Hence m^^ (w^k£^)1/s ■ ^^^-{w^t^ 1/r i^k and this completes the proof. D Remark (i) This result is due to Gleser, who gives similar results obtained by removing arbitrary sub-collections of a; [Gleser]. Remark (ii) Beckenbach, [Beckenbach 1964], has generalized Theorem 1 by assuming that m < di < Μ only for г such that 0 < no < г < n; this result is proved later, see IV 6 Corollary 3. Theorem 2 Assume that r, s £ R* and that α is an η-tuple that is not constant. If r is fixed, r > Q, there is a unique s = sq, Sq = r#, 0 < θ < 1, at which (Qn'sfe^)) attains its unique minimum value; if r < 0 there is a unique such So at which the unique maximum is attained. D Assume, without loss in generality, that the non-constant α is increasing, and put gr(s) = g(s) = log(Q^'s(a; w)) . Simple calculations give: // ч rYZ=1Wial log a,i 1 ^A 9{s)= Σ^^Ι logwr|>^ (η η η (Σ Wi<1i lQg2 tti) (Σ Wiui ) " (Σ WiaSi lQg °0' ■ - - - г=1 «=i i=i If then r > 0, (C) gives that д" > 0 and so g' is strictly increasing. Further ^oo^W = 4^^) <о; м5'(8) = 4£^) >* 9'{r) ΣΠ r by (J), and the concavity of the logarithmic function.

232 Chapter III These facts are sufficient to establish the result when r > 0; the case r < 0 is similar. D Remark (Hi) Some properties of gr,r > 0 are: there is a unique minimum at s = so, 0 < so < r; gr(r) = 0; if ro < s < sf then gr(s) < gr{s'). Remark (iv) Since Q^'s(a; w;)Qnrfe Ш) = 1 we can easily state a similar theorem with s fixed. Remark (v) From the Remark (ii) if 0<r < s, (Q%s(a;w))rs > (Q£r(a;w))r = 1, which is equivalent to (r;s). Remark (vi) Suppose that 1 < r < s then from the properties in Remark (ii), applied to 9l: (Q^{a;w))a > (QFfaw))*, or (su(a;uorr<5^^ ΣΠ r 1 i=i wiai a result that, in the equal weight case, can be found in [Berkolaiko]. An alternative proof has been given in [Mitrovic 1970). This is a particular case of the fundamental inequality between the Gini means, 5.2.1(7). Theorem 3 Assume that η > 2, α an η-tuple with 0 < m < α < Μ, w_ an η-tuple with Wn = l,r,sER,r<s, and let β = M/m, then 5(a;w)<Tr's(p), (2) where ΓΓ>3(β) = { Further if s-r β3 1/r β3 - βτ Τ 1/8 ifrs φ 0, β8 - βτ s ) \ s-r βτ -1 г.,.ОТ.л„г^).(^)"-«Р(^-1), „г.*. rlogPY/r /1 log/? ν·°(β)= lim s—►()+ ■~w-(^?) «*>■ /1 _ _log^_\ ifs = 0. (3) 0(r,s) = < s-rV/?r-l /?e-l/ ifrs^O, ifr = 0, (4)

Means and Their Inequalities 233 then 0 < 0(r, s) < 1, and equality occurs in (2) if and only if for some set of indices j} Wi = Θ, di = M,i e I and di = m,i <£ I. D Let us define rr's(/?) = sup{x; x = Q%s(a]w),Wn = 1,0 <m<a<M}, assuming, as will be shown, that this quantity does not depend on w. Writing Λ = {α; 1 < a < β}, W = {w; Wn = 1}, also define aeA Then using the basic properties of Q%s, Τ^(β) = sup Т^(р]Ш) = sup <&в(а;Ш). (5) wGW aeA,w_£W Clearly Tr,s(P) < β, and simple calculations establish the following identities: Γ''°(β)=Γ-°>-'(β), (6) (Γ^(β)Υ = Г^Ч/Г), 0<r< s< oo, (r°'s(/?))s = Г0'1^5), 0<s< oo, (7) (rr's(/?))~r = Γ"1'-β/Γ(^-Γ), -oo < r < 0 < s < oo. (8) Identity (6) shows that it is sufficient to evaluate Tr,s in the following three cases: (a) 0 < r < oo; (β) r = 0, 0 < s < oo; (7) -00 <r<0<s<oo. Further using (7) and (8) these three cases can be reduced respectively to a consideration of: (г) Г1'*, t > 1; (гг) Г0'1; (ш) Г"1'*, 0 < t (г) То evaluate Г1'*^) let us first consider Г1'*^,^)· Since the set Л is compact there is a b £ Л such that (г^,м))( = (Qi'fe))4 = ,£riWtt- (9) (Σ<=ι "Ά) η For some /с, 1 < /c < n, put for s = 1 or t, Wk = w, 2_\wi^l = (1 — w)as. Then if 1 < χ < β, the function φ, г=1 ,, ч ftntifu и и и ^^t ^ж* 4-(1 --ш)а4 0(ж)= (Q^i,··· Α-ι,ζΑ+ι,..·,&η;^)) = 7 Γν*' (гиж 4- (1 — w)a±) has a maximum either at χ = 1 or at χ = β; this is immediate from a consideration of φ'. Since /с, 1 < к < η, was arbitrary the point b of Λ given by (9) is such that

234 Chapter III hi — 1 or /?, 1 < % < n. Write / for the set of indices for which the second alternative occurs, of course / can be empty, and let у = Wj, when 0 < у < 1 and (Г^Ш))' = /~" + ^t = Ф(У), say. (l-y + yp) By (5) (Γ^(β))τ = sup0<?/<1 ф(у), and simple calculations with ip' show that φ has a maximum at yo, 0 < yo = #(1, t) < 1, where θ is defined by (4). Hence i" is not empty and Γ1'*(/?) = ( ψ(θ(1, у)) ) , which is the value given by (3). The cases of equality are immediate from this argument. This proves the theorem when either —oo<r<s<0or0<r<s<oo. (гг) As in (г) there is a b e Л such that г°»1(Аш) = 0^1(Ь;ш) (exP ΣΓ=ι ™* loS Μ For some к, 1 < к < η, put Wk = W, jyi=i Wit ^ г^к Then if 1 < χ < β the function φ, Wk = w, jy!=i Wibi = (1 - w)ai, Σ^=1 ^ibgbi = (1 - w)ao. г^к г^ак ., ч гуж4- (1 - ΐϋ)αι 0(ж) - u> log ж 4- (1 — гу) log αο' has a maximum either at χ — 1 or at χ = β. The argument then proceeds as in (г), and gives the theorem when either —oo <r<s = 0or0 = r<s<oo. (ггг) As in (г), or (гг), there is a b £ Л such that (r-1-i(/3,«;))t = (E^)(E^^1)t. i—i i=i For some /с, 1 < /с < η, put η η wk=w1 ^Y^Wib* = (1-w)at and ^ги^"1 = (1 - w)a_i. г=1 г=1 Then if 1 < χ < β the function φ, φ(χ) = (wxb 4- (1 — ги)а*) (wrr-1 4- (1 — w)a-i) has a maximum either at χ = 1 or at χ = β. The argument then proceeds as in (г) and gives the remaining case of the theorem, —oo<r<0<s<oo. D

Means and Their Inequalities 235 Remark (vii) The above result is due to Specht, and was later rediscovered by Cargo & Shisha who gave the cases of equality; [Cargo h Shisha 1962; Specht]. An alternative proof of Specht's result is in the paper by Beckenbach, [Beckenbach 1964). See also [Laohakosol Sz Ubolsri). Another proof of Theorem 3 has been given by Goldmann, based on the following inequality, that in the case r = s = 1 is due to Rennie, see [Goldman; Rennie 1963). Lemma 4 If -oo < r < s < oo, rs <Q and 0 < m < α < Μ then (Ms - ms)(m^]{a;w))r - (Mr - mr)(m^](a;w))s < Msmr - Mrms. (10) If rs > 0 then (~W) holds. D This lemma can be deduced from I 4.4 Theorem 23; see [Beesack Sz Pecaric; Pecaric L· Beesack 1987b). D To deduce (2) from this lemma rewrite (10) as r Mr -mr fm[n](a;w)\s s Ms - ms (OTl^1 (a; w) V Msmr - Mrms s — r rmr \ m J s — r sms \ m J (s — r)mrms and use (GA) to get iMT -mr /Э745](а;ш)ч Л ~r^s~^ /Ms - ms /OT^fej^ ^ s/(s"r) \ rmr V m J J \ sms V m / Msmr - Mrms {s — r)mrms This, on rewriting, is (2). Remark (viii) Theorem 3 in the special case r = 1, s > 1 was considered earlier; see [Knopp 1935). Other references are: [Mitnnovic & Vasic p. 78), [Cargo 1969; Diaz, Goldman L· Metcalf; Diaz L· Metcalfl963; Lupas 1972; Marshall Sz Olkin 1964; Mond L· Shisha 1965; Newman Μ A; Rosenbloom; Shisha Sz Mond; Wang Sz Yang; Zhang Ζ Η). The following more classical inequalities are special cases of Theorem 3; [Kan- torovic; Schweitzer P). THEOREM 5 (a) [Kantorovic's Inequality] If a, w_ are η-tuples, η > 2, with Wn = 1 and 0 < m < α < Μ, then Е^ДЕ^-щ^-. (11) 2 = 1 / \ = S1 l /

236 Chapter III with equality if for some index set I, Wj = 1/2, a^ = M, г £ I, and a^ = m,i fi I. (b) [Schweitzer's Inequality] If α is an η-tuple, n>2, 0<ra<a< М, then n2(M + m)2 i> ς^^^. (». a,i) AMm with equality if and only if there is an index set I containing n/2 elements, and di = M, г Ε I, and a\ = га, г φ I; in particular (12) is strict if η is odd. D (a) We give four proofs of this result. (г) Put r = —1, s = 1 in Theorem 3. (гг) [Henrici] Determine two η-tuples α, β as follows: di = αιΜ 4- β%τη, α"1 = агМ~г 4- β%νη~γ, 1 < г < п. It is easily seen that both of these η tuples are non-negative, and further -1 = (a, + A)2 + aiP^M^J ; (13) so in particular а 4- β < е. Now let A = Σ™=1 iDiOti and Β = Σ™=1 ^ΐβΐ when η A + B = YjWi{ai+pi)<Wn = \. (14) Hence the left-hand side of (11) is equal to {AM + Bm^AM-1 + Вт-1) ={A + B)2 + AB ^M ~ ^ v ; AMm Шт This completes the proof of (11). There is a equality if and only if A 4- В = 1 and, using the case of equality in (GA), A = B. From (14) the first condition implies that a + β = e, which from (13) implies that for 1 < ί < η either oti = 0 or βί = 0; equivalently a; = Μ or m. So if / is the set of indices for which сц = Μ the condition A = В implies that Wi = 1/2. (in) [Ptak] г=1 г=1 г=1 ν г=1 4(к+Ш *Ι2-2<23)-

Means and Their Inequalities 237 (iv) It is possible to deduce (11) from (12). We can without loss in generality assume that that wi = vi/V,V,Vi e N*, 1 < i < n, Vn = V. Then simple calculations show that (11) reduces to a case of (12). (b) This is just a special case of (a). D Remark (ix) Proof (гг) is based on a method used in [Polya & Szego 1972 p. 71} to prove inequality (20) below. Remark (x) The observation that (12) implies the more general (11) is due to Henrici; see [Henrici]. Remark (xi) Theorem 3 has been used to obtain a matrix inequality that generalizes Kantorovic's inequality; see [Mond 1965a] and VI 5. Remark (xii) Kantorovic's inequality has been obtained in the form 21η(α)-ϋη(α) < (M-m)2 ΐη(α) ~~ 4Mra ' by Weiler, who then points out that because of (GA) the same inequality holds with the harmonic mean replaced by the geometric mean; [Weiler]. A completely different type of converse inequality has been proved by Rahmail; see [Pecaric 1983b; Rahmail 1978; Vasic L· Milovanovic]. Theorem 6 If 0 < r < s, w_ an η-tuple, α a monotonic, concave η-tuple, and η = (0,1, 2,..., η — 1), η = (η — 1, η — 2,..., 1), then ^:\а-ш)<аШ^(а;Ш); (14) where if α is increasing а = т^(тш)/т[:](п;ш); (15) while if α is decreasing, a = m[°\n;w)/m[r\n;w). If a is convex with a\ = 0 and 0 < s < r then (~14) holds with α given by (15). Remark (xiii) The proof uses the monotonicity of the n-tuples -—г—, 1 < г < η, and , 1 < г < η — 1. η — г Remark (xiv) The second part of the theorem has been generalized to /c-convex η-tuples; see [Milovanovic L· Milovanovic 1979]. The following result is related to those in 2.5.4 and to II 3.8 Theorem 29.

238 Chapter III THEOREM 7 If the η-tuples a, b, b/a are either all increasing, or all decreasing, and ifr<s then f(x) = Q£s((l - x)b + χα; w), 0 < χ < 1, is decreasing. In particular Q£s (b; w) > Q£s (a; ιυ) . Remark (xv) In the special case r = 0, s = 1 and α constant this is due to Pecaric; the general result was proved by Alzer; [Alzer 1990i; Pecaric 1984d]. 4.2 Differences of Power Means An upper bound for D)^s(a; w_) was given by Mond & Shisha, and is Theorem 10 below; [Mond L· Shisha 1967a,b; Rosenbloom; Shisha h Mond] We first prove two lemmas. Lemma 8 IfO<m<x<M and r < s deune f(x) = r(xr — axs — β) where Mr-mr л Msmr-Mrms r, ч л a = — , β = — , then fix) > 0, m < χ < Μ. Ms - ras' H Ms -ms ' J v ) D Simple computations show that if we consider f(x), χ > 0, then f has a unique zero. Since f(m) = f(M) = 0 all we need to show is that f'{m) > 0. However putting у = Μ/га, f'{m) = rm?-1^ — asms~r) = rg{y)/{ys — 1) and so sgn/'(ra) = sgn(rs)sgnp(y). Noting that g(l) = 0 and g'(y) = rsyr-1{ys~r - 1) and that g'{y) > 0 if rs > 0, g'{y) < 0 if rs < 0, completes the proof. D Lemma 9 With the above notations deune the function h by h(x) on the interval J, xxls -(αχ + β)1^, ifrs^O, 1/« f-к r ι \(x—ms)/(Ms—ms) .r _ x1/s -m(M/my ' \ ifr = 0, fn/fl \(x-mr)/(Mr-mr) ц {m{M/m) ' -xL/r, ifs = Q, ( [min{Ms,ms},max{Ms,ms}], if s φ 0, ~{[Mr,mr], ifs = 0. Then there is a unique у eJ where h takes its maximum value. Further ( ((ΐ-θ)πι8 +6Ms)1/s - ((l-6>)mr + 6>Mr)1/r, if rs φ 0, < ((\-θ)πι8 + ΘΜ8)1/8 -т}-вМв, if r = 0, [ mx-QMQ - ((1 - в)тг + 9Mr)l/r, if s = 0, h(y) where у —m° Ms - ras' у — mr Mr - ras' ifs^O, ifs = 0.

Means and Their Inequalities 239 D Let us consider the case rs ^ 0. о Since h(ms) = h(Ms) = 0 and, by Lemma 8, h(x) > 0,ж Ε J, it is clear that for о at least one у £ J and h'{y) = 0. Suppose that у is not unique; that is there are two points yi, 2/2> 2/i < У2, such that h takes its maximum value at both. Now if h'(x) = 0 then h»{x) = l xV-\ax£ - -) + β{~ - 1)). Since h"{yi) < 0, г = 1, 2, it follows from the above that for г = 1,2 we must have that -(ayi( )+/?(--l)) < 0. s s r' s ' Now for some x, yi < χ < У2 , h'{x) = 0 and this is a local minimum of h. However from the expression for h" we see that h"{x) < 0. This is a contradiction, so у is unique. The cases r = 0, or s = 0, can be discussed in a similar manner, and the rest of the lemma is a result of simple calculations. D Theorem 10 Assume that η > 2, α an η-tuple with 0<m<a<M,w_an η-tuple, r, 5 e R, r < s, and α, β, h, y, θ are as in Lemmas 8, 9 then v>rris(u;m)<h(y) (16) with equality if and only if for some index set I, Wj = θ,α,ΐ = M,i ζ Ι,α,ί = m,i fi I. D (i) rs ^ 0 Applying Lemma 8, η η and so ЩЧ&;ш) < Vrf:\a;w) - (a(Tt\a;w)Y + β)1/Γ = h^Tl^^w))3), (17) with equality if and only if for each г, 1 < i < η, α^ = m or M. This by Lemma 9 gives (16), with equality if and only if for each г, 1 < г < η, α^ = πι or M, and Ση s After simple computations this completes the proof in this case.

240 Chapter III (ii) If instead of using Lemma 8 we note that if m < χ < Μ, q > 0 then, by the strict concavity of the logarithmic function, xq > Mq{x3-m3)/{M3-rn3)mq{M3-x3)/{M3-m3)^ with equality if and only if χ = Μ or πι, then we can obtain (16) in the case r = 0, or s = 0. This completes the proof. D Remark (i) The case r = 1, s = 0 was considered earlier; see [Knopp 1935]. Remark (ii) In [Cargo Sz Shisha 1962] it is shown that the maximum of D)^s, and also of Q^'s, occurs at a vertex of the η-cube; see also [Pecaric L· Mesihovic 1993]. Remark (iii) The methods used to obtain converses of (J), I 4.4 Theorem 28, and (GA), II 4.1 Theorem 1 proof (ii) can also be used here; see [Bullen 1994a]. 4.3 Converses of the Cauchy, Holder and Minkowski Inequalities Converse inequalities for (C), (H) and (M) can be obtained as corollaries of the results obtained in sections 4.1 and 4.2. Theorem 11 Let b and с he η-tuples, peR, ρ > 1, 0 < m < Μ, β = M/m, and suppose that, m < b1/p' /c1^ < M, then (ΣΙ^Γ)1/Ρ(ΣΙ^')1/Ρ' < //р-/?Л/ ρ' γ/*, ρ γ" гш (Σ?=Λ*) -ν ρ+ρ> )\ι-β-ν) \βρ-ι) ■ liej 1 / ρ' Ρ λ Further if θ = ( —7 — — then equality occurs if and only if there ρ -f ρ' V 1 — β-Ρ βΡ — 1 / is a set of indices I such that, (1 — ^)Y^iejbiCi = O^^jbiCi and b/p c~ 'v = M, г e I, = m, г φ Ι. If 0 < ρ < 1 then (~18) holds. D If ρ > 1 this is an immediate consequence of 4.1 Theorem 3 by putting s = p, r = -ρ'', α = b1/p с'1/? and Wi = biCi/ Y%=1 bkck, 1 < г < п. The other cases can be found in [Beesack Sz Pecaric; Vasic L· Pecaric 1983]. D Remark (i) ltb<b<B,c<c<C then the right-hand side of (18) can be replaced by the simpler (B/b) ip/c) ; see [Crstici, Dragomir L· Neagu]. Other results have been given in [Barnes 1970; Laohakosol L· Leerawat]. A particular case of (18) is the following result; [Watson; Greub L· Rheinboldt; Masuyama 1982,1985].

Means and Their Inequalities 241 THEOREM 12 [Cassel's Inequality] Ifb,c,ui are η-tuples with 0 < mi < b < M\, 0 < Ш2 < с < M2, where mim2 < M\M2 then i=i i = i i=i Remark (ii) The case of constant w. can be written and is called the Polya-Szego inequality; see [Polyа & Szego 1972 p.71). Clearly (19) and (20) are equivalent. Remark (iii) Inequality (20) is a special case of Kantorovic's inequality, 4.1 (11); conversely, as Kantorovic pointed out, (11) follows from (20). For a generalization of (20) see [Chen Υ L] Remark (iv) All of the inequalities (11), (19) and (20) can be deduced from an integral form of Schweitzer's inequality, VI 1.3.1(11); [Alp.60, Remark 2], [Makai]. An alternative proof of these results has been given by Diaz & Metcalf that is based on the case ρ = 2 of the following theorem; [PPT p. 115], [Diaz &; Metcalf 1963 1964a, 1965; Mond L· Shisha 1967b; Shisha L· Mond]. Theorem 13 If w_,b and с are n- tuples, ρ > 1, and if 0 < m < c}fv'b~1/v < Μ then η η η {Μ"' - тр')^ю^ + (тр'МРр'-Мр'трр')^ю^' < {Мрр'- трр') ^шД^, г=1 г=1 i=i (21) with equality if and only if for all i, 1 < г < η, c/p b~ — m or Μ. The same inequality holds if ρ < 0 but if 0 < ρ < 1 then (~21) holds. D This is a consequence of I 4.4 Theorem 23; see [PPT pp. 1Ц-115]. D Let us now see how inequality (20) follows by taking equal weights and ρ = 2 in (21). Put m2 = m2/Mb M2 = M2/mi in this special case of (21) to get Σ 2 , rn2M2 ^ 2 (M2 ™>2\x^h i=i i=i i=i

242 Chapter III V 777.1 Μι ) ί-J V mi Mi J i ГП2М2 тгМг η η Σ*? Σ* г=1 г=1 > ΛΙ^^-ΛΙ mi Μι ^ г ) ~ This on rewriting gives (20). A converse of (M) can be deduced from the converse of (H) given in Theorem 11; [Mond L· Shisha 1967b]. Theorem 14 Let b,c,m,M,P,6,p be denned as in Theorem 11 but now with m < Ь1/р'(Ь + с)1/У, с^Р'^Ъ + с)1/*' < M; then l/p (ЕГ=1ьГГР + (Е1и vVP <E; where Ε is the right-hand side of (18). There is equality if and only if: (i) there is a set of indices, I, such that (1 - 0) $>(b< + c^-1 = O^biibi + d)*-1, iei φ with {pi/{hi + Ci)) — Μ, or m according as i Ε I, or i <£ I; and, (ii) there is a set of indices, J, such that with (α/^ + α))1/ρ' =M, or m according asiEJ,ori<£J. If either 0 < ρ < 1, or ρ <Q ,then (~24) holds. D If ρ > 1 then by Theorem 11 η η η ^(Ьг+ъг =j2bi(-bi+c')p_1+Σ^(^ + c*)v~1 i=i i=i i=i i=i i=i + Е-\±(*)1/р(£(Ь,+с>У)1/р', which gives the result in this case. The other cases have similar proofs. (23) D

Means and Their Inequalities 243 Theorem 15 Let b and с be η-tuples, ρ > 1, 0 < m < b1/pc1/p < M; then with the notation of 4.2 Lemma 9, with equality if and only if there is an index set, I, such that (1 — Θ) ^2ieI hci = θ Σΐέΐ biCi, &яс/ bi c~ lv — M, or m, according as г £ I, or i <£ I, D This follows from 4.2 Theorem 10 in a manner similar to the way in which Theorem 11 follows from 4.1 Theorem 3. D Remark (v) In the cases ρ = 1,2 converse inequalities of a different type have been obtained by Benedetti who considers the maximum possible value when the η-tuples have terms that are restricted to certain finite sets of values. However it is beyond the scope of this work to consider the converses of (C), (H) and (M) in more detail. The reader is referred to the standard works [ΑΙ; ΒΒ; PPT] for further references in this area; see also [Dragomir L· Goh 1997b; Izumino; Izumino & Tominaga; Toth]. Remark (vi) Converse inequalities have been based on the order relation of I 3.3; see [Pecaric 1984a]. Remark (vii) A converse of the equal weight case of 3.1.3 (9) has been given by Toyama; see [AIp.285], [Toyama]. ОТМ(ЯЛИ(%))) < (πιίηίτη,η})1/"^1^^'))· This was given a simpler proof and extended to the weighted case in [Alzer &: Ruscheweyh 2000]. Remark (viii) Leindler has proved that if 1 < p,q,r < oo, and if 1/p + 1/q = 1 + 1/r then oo oo oo oo ( Σ <)1/ρ( Σ b,n)1/q< Σ ( Σ («-™)1/r; n= — oo n= — oo n= — oo m= — oo see [Leindler, 1972a,b,c, 1973a,b, 1976; Uhrin 1975]. The next result is a discrete form of a result of Zagier; [Alzer 1992e; Pecaric 1995b; Zagier).

244 Chapter III Theorem 16 Let d,b,c,d be sequences, a and b decreasing to zero, c,d< I and let A = γ^χ α», Β = γ^Ζλ Ь», С = J^ Ci,D = ΣΖι di then OO V^OO V^°° J L l^ait)i- тах{С7,т ' i=i k J In particular if a,b < 1 ^ тах{А,Б} D For any j > 1, OO OO j / v a^Ci = Cdj + 2^(«г — CLj)Ci < Cdj + ^(^г — CLj)Ci, i=i i=i i=i SO oo j Dj V^diCi < jCdj + D 2_](ai ~~ аз) — max{C> D}Aj. i = i i=i Hence OO AbjDj ^2 аг°г ^ max{C, D}AjAbj. which on summing over j gives the required result. D Remark (ix) Obviously from (C), OO OO OO ΣαΑ<™η{Α,Β,(Σα*)1/4Σΰ)1/2}- i=i i=i г=1 The following result is in [McLaughlin]. Theorem 17 Ifd,b are real 2n-tuples then η 2η 2η η (Х]а2^2г-1 -a2i-lb2i) < 2^а?Х^? ~~ (X]a*^) · г=1 г=1 An inequality that is a mixture of (H) and (M) has been proved in [Iusem, Isnard L· Butnariu]. Theorem 18 Ifd and b are two η-tuples, ρ > 2, deune Ci = Щ~ , 1 < г < η, then г=1 г=1 i=i г=1 i=i i = i

Means and Their Inequalities 245 If 1 < ρ < 2 the opposite inequality holds. 5 Other Means Defined Using Powers There are many other generalizations of the classical arithmetic, geometric and harmonic means besides the power means. Some generalizations are based on the very close connection of the power means with the convexity of certain functions; such generalizations are taken up in the next chapter. Other generalizations are really only defined in the case η = 2 and these are considered in VI 2. Here we study some generalizations that like the power means are based on the use powers, logarithms and exponentials. 5.1 Counter-harmonic Means Definition 1 If α and ш are η-tuples and r Ε Ш then the r-th counter-harmonic mean of α with weight ш, is С](а;ш)={ ΣΠ r '-^ ifr е. Wja- r— ι ' Ση max a, ifr — oo, mina, ifr = —oo. As with previous means we will just write S)^(a;w_) when η = 2, S)n if the reference is unambiguous, S)%} (a) will denote the equal weight case, and if I is an index set the notation S)[ (a;w) is used in the manner of I 4.2, II 3.2.2. The following identities are easily obtained: !ry ' ^(a, b) = Φ(а, Ь), and fin](a;w) = 2U(а;ш); ^L°](а;w) = Яп(а;ш); Remark (i) The reader can easily check that i)JJ(a) is the point at which the function Σ™-ι {(си ~ x)/x) takes its minimum value. Theorem 2 (a) If 1 < r < oo then $](аж)>Я$](а;ш), (2) and if —oo < r < 1 then (~ 2) holds. Inequality(2) is strict unless r = l,oo or а is constant. (b) If —oo < r < 0 then the following stronger result holds: ^](a-w)<Tt-1](a;w). (3)

246 Chapter III Inequality (3) is strict unless r = — oo, 0 or α is constant. □ The cases r = ±oo are trivial so assume that rel; then If 1 < r < oo then (r;s) and (4) imply (2), while if — oo < r < 1 (~2) is implied. This completes the proof of (a). Now assume that r < 0 then: й1г1(«;ш) = (4-г+11(й-1;ш)Г1 < (©t^V1;™))-1, by(a), = апкг"ч(а;ш)· This gives (b), and the cases of equality are immediate. D Remark (ii) Inequalities (2) and (3), together with 1 Theorem 2(c) justify the cases r = ±oo of Definition 1; see [HLP p. 62]. Remark (iii) Inequality (2) in the equal weight case and with r = 2 is due to Jacob; [Jacob]. Theorem 3 If α and w_ are n-tuples and if — oo < r < s < oo then Ъ1;](а;ш)<Ъ[:](аЖ), (5) with equality if and only if α is constant . D (г) By 2.3 Lemma 6(b) and 2.3 Remark (iii), ra(r) = Σ™=1 Ίΐ)%α\ is strictly log-convex if α is not constant; see also 3.1.1. So by I 4.1(3), if —oo < r < s < oo, log om(r) — log om(r — 1) < log om(s) — log om(s — 1); which is just (5) in this case. If either r = —oo or r = oo the result is immediate, (гг) In the case that (s — r) Ε Ν the following proof has been given; see [Angelescu]. First note that m(r — l)x2 — 2m(r)x + m(r + 1) = Σ^=1 ^(ж — α^2α\~γ. Hence this quadratic does not have any real zeros; so m2(r) < m(r + l)m(r — 1), which implies (5) with s replaced by r + 1. D It is easy to check that these means have the properties (Ho) and (Co), and (5) shows that they are strictly internal. However counter-harmonic means do not have the properties of (Mo) and substitution. Consider the following examples; [Beckenbach 1950; Farnsworth L· Orr 1986].

Means and Their Inequalities 247 Example (i) If a = 1, b = 2, с = 3 then S)f (1,2,3) = 7/3, S)f](l,2) = 5/3 but &ψ (5/3, 5/3, 3) = 131/57 φ 7/3 Example (ii) Consider h(x) = ϊ)ψ(χ, 1) = (l+x2)/(l + x) ; then h(0) = 1 = h(l) and h has a minimum of 2(л/2 — 1) at ж = л/2 — 1. Example (in) Consider χ(α) = йп fe) then Vx(a) = —ЛггК"1 + (1 - Γ)α-2χ(α)), and so ί)^ (α) is monotonic if 0 < r < 1. Theorem 4 If 1 < r < 2 then £lr] (a + 6; w) < fiM (a; ™) + S)$ (b; щ); (6) if 0 < r < 1 then (~ 6) holds. Inequality (6) is strict unless either r = 1 or a ^ b. D The case r = 1 is trivial so assume r > 1; then by Radon's inequality, 2.1 (9), with η = 2, ρ = r we get that ((ΣΓ=1^<)1/Γ + (ΣΓ=1^)1/Γ)Γ v y -<^](й;ш) + ^г1(Ь;ш). )7 Then (6) follows using (M) on the numerator of the left-hand side, and (~ M) on the denominator. The case of equality follows from the cases of equality in the inequalities used in the proof. The proof when r < 1 is similar. D Theorem 4 is trivial if r = ±oo but if2<r<oo, or — со < г <0 then counterexamples show that the theorem may fail. Example (iv) If 2 < r < oo take α = {1, e, e,..., e} and b = {1, e2, e2,..., e2} for a suitable positive e. Remark (iv) These means and their properties seem to be due to Beckenbach; [Beckenbach 1950) where an alternative proof of (6) can be found. As a result (6) is often called Beckenbach^ inequality; another inequality with the same name is given above in 2.5.5; see also [BB p.27], [Bellman 1957b]. The above proof is

248 Chapter III due to Danskin; [Danskin]. Other references are [Bagchi L· Maity; Bellman 1954; Brenner & Mays; Castellano 1948; Lehmer; Wang W L & Wang Ρ F 1987]. 5.2 Generalizations of the Counter-harmonic Means Various authors have generalized the ratio in 5.1 (1) and while not obtaining such detailed results have extended Theorem 3. Mitrinovic & Vasic considered Γ(χ)_ Т1[°](ах;ш) showing that if r = s then F is increasing, or decreasing, according as rk > 0, or rk < 0; if s < r then F has a unique maximum at χ = rk/'(r — s), whereas the same point is the unique minimum if s > r. The methods of proof are those of elementary calculus; [Mitrinovic &: Vasic 1966a]. The equal weight case with r = s had been considered earlier; see [Izumi, Kobayashi L· Takahashi; Kobayashi]. Liu & Chen, [Liu L· Chen], have considered the ratio «fete))*"1' which if s = t = r and, r — 1 is substituted for r is just (1), in the case of r finite. They then generalize (6) as follows. Theorem 5 Ut > 1, and s>l>r then £H(a-f b,ur,r, s;t) < £H(a, w;; r, s; £) 4- ЩЬ,Ш.',г, s;t). while ift < 1, and s < 1 < r the opposite inequality holds. Other authors have used ratios similar to those in the previous section to define new means; [Castellano 1948; Dresher; Gini 1938; Gini L· Zappa; Godunova 1967; Ku, Ku L· Zhang 1999; Pales 1981; Pompilj]. 5.2.1 Gini Means Let α and w_ be η-tuples and for p,q £Ж define the Gini mean of α with weight w_ by ρ\1/ΣΓ-ιΐϋία? Π η WiaK \ ^-ο-ι Convention Since (3^q = Q5^p we will assume in this section that ρ>q .

Means and Their Inequalities 249 The Gini means include both the power means and the counter-harmonic means as special cases as the following simple identities show. 0£'°(а;ш) = a«M(o;te); в^-Ча-ш) = S$(a;w). The means &+1'(a,b) =#+1)(a,6) = br + a*r+ ,re R, have been called Lehmer means] [Alzer 1988c; Lehmer]. When r = 1 the Lehmer mean is called the contraharmonic mean6; [2], ,s b2 + d2 0J'1(a>b)=UM(e,b) = Ь + a ' Some properties of Gini means are given in the next theorem. Theorem 6 (a) птвГ(«;ш) = 0Г(«;ш); p^q lim 0^'9(a;w) = max α; lim 0^'9(a;w) = mina. р—кх) q—> — oo (b) Ifpi < P2, <?i < <?2 then further if either pi ^ p<i or qi =^ 172 then inequality (7) is strict unless a, is constant. (c) If ρ > 1 > q > 0 then <ЗГ(а + &Ш) < ^'9(а;ш) + 0™(&*>). (8) D (a) The first limit is a simple use, after taking logarithms, of l'Hopital's Rule7. As to the second: log(д™ в»*&^) %^^(μΣ><) ~1οε(Σ>α?)) 1 oo = lim - \og(S^ Wid?), p—юо Ό z—' = log (max α), by 1 Theorem 2(c). The contraharmonic mean is the solution χ of the proportion x — a:b—x::b:a and is a neo-Pythagorean mean; see VI 2.1 4 7 See 1 Footnote 1.

250 Chapter III The third limit is handled in a similar manner. (b) It is easily seen that if ρ =fi q logoerfe») = logom(p)-logow4 P- Q where m is the function defined above in the proof of 5.1 Theorem 3, or in 3.1.1. This function is log-convex and so (7) in the case pi =£ qi and p2 ¥" 42 is immediate from a basic property of convex functions, I 4.1 Remark (v), and the fact that the logarithmic function is strictly increasing; see [PPT pp.119-120]. The remaining cases follow by taking limits. (c) Assume that ρ > q > 0 and write the right-hand side of (8) as, К'я(а;ш) + К'Чкш) = " w- „,,„_„, + Now apply Radon's inequality, 2.1 (9), in the case η = 2, and ρ in that reference taken as p/(p — q) to get лм, ^^,nh ^ №Ча;ш) + ж%](Ъ;ш))р/{р-я) _ иы(а;ш) +anh](fcjji))«/CP-«) Now by (Μ), ρ > 1, and (~M), 0 < ς < 1, OTl|fi (а; щ) + 9Jtjf1 (b; w) > SPtJT1 (o + fe щ) απ[?] (a; m) + ®4?] (& ш) _ απί?] (α+& ω)' Hence The other cases of (8) are easily proved. D Remark (i) The proof of (b) is due to Pecaric & Beesack; [PPT p. 119], [Pecaric &Beesack 1986]. Remark (ii) The case pi = 1,<?ι = 0 of (7) has already been proved; see 4.1 Remark (v). Remark (iii) Inequality (8) was first proved by Dresher and is sometimes referred to as Dresner's inequality. The proof, which generalizes that of 5.1 (6), is due to Danskin; [PPT pp. 120-121], [Danskin; Dresher]. A very detailed examination of

Means and Their Inequalities 251 the case η = 2 has been made in [Losonczi L· Pales 1996]. See also [B2 pp.233, 264], [Guljas, Pearce L· Pecaric]. An extension of (8) is given in IV 7.1 Corollary 5. Remark (iv) Note that Liapunov's inequality, 2.1 (8), is just ^*(а;Ш)<в;'*(а;ш), 0<t<s<r. Remark (v) These means have been generalized to allow for complex conjugate ρ and q, [Pales 1989]. For another generalization see V 7.3. Gini means have been studied in detail by many authors; see for instance [Aczel & Daroczy; Allasia 1974-1975; Allasia & Sapelli; Brenner 1978; Clausing 1981; Daroczy L· Losonczi; Daroczy L· Pales 1980; Farnsworth L· Orr 1986; Jecklin 1948b; Jecklin L· Eisenring; Losonczi 1971a,c 1977; Pales 1981, 1983a, 1988c; Persson & Sjostrand; Stolarsky 1996]. 5.2.2 Bonferroni Means Another kind of generalization has been suggested by Bonferroni, the Bonferroni means; [Bonferroni 1926, 1950]. Let α be an η-tuple and define i/(p+g) 1 »£*(*) - τττ^ L· *?*■ n(n- 1) ^ 3 L with obvious extensions to *8£'9'г(а) etc. Theorem 6 With the above notation ifh>0,q<p — h < p then »Г(й) > ®rM+h(fi)· D For a proof the reader is referred to the references. D 5.2.3 Generalised Power Means In a very interesting paper Ku, Ku & Zhang have considered a very general extension of power means ; [Ku, Ku L· Zhang 1999]. Let ki, 1 < г < га, be distinct positive functions defined on the η-tuples a, and write k_= (/ci,..., km). If ;ш, х_ are га-tuples, x_ allowed to be non-negative, and such that Σ™-± WiXi = 1. If r £ R the r-th generalized power mean of α is : fi№ntekm,x)=\ (S^^i(*i(fi))r)1/r. if ^°. The generalized power means include power means, certain Gini means, in particular the counter-harmonic means.

252 Chapter III Example (i) Take ki(a) = сц and xi = W~x, 1 < г < η, when Л^ n(α; 4.; w_, ж) = ^n (й;ш)· In fac^, as we easily see, ЛЙ,„(а;&ш,2) =Ш1Й(*1(а)."-.*т(о);Зйг)· (9) Example (ii) More generally take ж^ = ^n г «, /сДа) = сц, 1 < г < η, when we find that Я^"*1 (α; fc;ш,ж) = ®[ηΜ{α,ш),Р^Ч- То obtain an extension of (r;s) here we need two further concepts. Firstly we define an order on the set of ж; x_^> x_' <=> there is a го, 1 < го < ra, such that Xi >x'n 1 < г < г0, Яг0+1 < ж-0+ь Xi < x'i, г0 -f 2 < г < га, if г0 4- 2 < га. Secondly we define a condition of equality for the k: EQ(k,x >> ж', α) holds if and only if ra У^ (/ci(a) — ki0(a)) = 0 <=>· α is constant. г=1 Example (iii) If for some distinct г, j we have /сДа) = kj(a) <=> a is constant then EQ(k,ai >> ж', а) holds for all distinct ж, ж', with ж >> ж'. Example (iv) Particular cases of Example (iii) are given by: ki(a) = UJl%i](a;,w), and ki(a) = (αη(ο))(η~Γι)/Γι(η"ι)(β„(ο))η(Γι_ι)/Γ'(η~ι)) for 1 < i < m, and where 0 < ri < · · · < rm <n. Theorem 7 (a) Ifr,seR,r<s, then &m,nfe & Ш,») < Я^ n(α; /с; w, ж), with equality if and only if α is constant. (b) If к is decreasing, that is /сДа) > &ΐ+ι(α), 1<г<га, r>0, and ж ^> ж' then <n (ffi & Ш, S) > С (a; fc; ш, г') · (Ю) If ж т^ ж' and ifEQ(k, ж ^> ж', α) is satisfied then (10) is strict unless α is constant. D (a) This is immediate from (r;s) and (9).

Means and Their Inequalities 253 (b) (г) r = 0. Choose a β so that pkm > 1. Let го be the suffix given in ж > ж', then since к is decreasing, (pki)WiXi = (pki)Wi<(pki)Wiixi-x^ > (pki)WiX'^pkio)w^Xi-x^ 1 < i < i0. Further Y™=x WiXi = ]Γ™ ι w^ = 1 and so ^=1 w^-x^ = 5Xi0+i ™»(я--ж»). Using these facts we get that го τη Λ1η(α;Λ;ω,χ) = Π(№Γ"(α) Π (№)"iXi(«) г=1 г=го + 1 >(^(о))^ш,("-1')Ц05к(Г!(а) Π GW'te) i=i г=го + 1 г=1 г=го + 1 т io m > Π (№)""(*ί-"")(α)Π(№),,"*1(α) J] (/^"""(α) г=го + 1 г=1 г=го + 1 га =/?ПСЖ%) =/?<„(a;fc;w>£')· г=1 Equality holds if for г ^ г0, 1 < г < m, (/^^^"^(а) = (/ЗА^)"7*^*-^). Since x^> χ' there is some г such that жг· =£ a^ and so the condition in the last sentence implies that if г =£ г'о then /сДа) = /c^0 (α), and so by the condition EQ(k, x^>x',a), we must have that a is constant. (гг) г > 0 As the proof of this case is analogous to the one above it will be omitted. The reader is referred to the reference for more details. D Remark (i) Particular cases of inequality (10) are inequalities 5.1(2),(5) and inequality 5.2.1 (7). Example (v) Let m = 3 and ki(a) = 2ln(a), k2(a) = <$n(a), &з(a) = S)n(a) and take r ^ 0 in (10). If χ > ж' then (ΐϋιχι2ΐ;(α) + г^2ж2^п(а) + ^3ж3Йп(а))1/Г > (^«(а) +w2x'2ern{a) + ™3«(θ))1/Γ- The quantity here, #3 n [а;/с; ;ш, ж], can be considered as a two parameter family of means lying between the extremes 2ln(a) and fin(a). 5.3 Mixed Means Let α be an η-tuple, к and integer, 1 < к < η, and denote by щ ,. · ·, orK the κ,κ= (^), /c-tuples that can be formed from the elements of a.

254 Chapter III Definition 8 If s, t e Ш then the mixed mean of order s and t of a taken к at a time is Wln(s,t;k;a) = Я«М (Я^1 (o^), 1 < i < к). (11) Example (i) The following special cases are immediate: mn(s,t;l-a) = mn(s,s;k]a)=m^](a); mn(s,t]n-a) = Tl^(a). Remark (i) An immediate consequence of (r;s) is that 9Яп(л £; &; &) is an increasing function of both s and £. The main results of this section are due to Carlson, Meany and Nelson; [Carlson 1970a,b; Carlson, Meany & Nelson 1971a,b]. Theorem 9 If -co < s < t < oo then Wln(s,t',k- l;a) < 9Jln(s, t; k; a). (12) If s > t then (~ 12) holds, and there is equality in both cases if and only if α is constant. D Denote by a·; , 1 < j < /c, the collection of (k — l)-tuples formed from the elements of a\ '. Then each orh~ , 1 < h < к' = (fc^1), occurs (n — k + 1) times in the collection of «L > 1 <J < /c,l <г</<с; note that (n-fc + 1). Firstly and so if s < t we have by (r;s) that ш#(а*) >mf(т^а^),i<j<k). (is) Hence, by (11), and using the above notation, VJln(s, t; k;a) >9J4S] (τΐ[°] {m[tMj), 1 < J < *), 1 < » < «) =апМ(ал111(а*-1),1<л<к') =Wln(s,t; k — l;a). If s > t the inequality (13) is reversed. The cases of equality are immediate. D We now wish to obtain an inequality between different mixed means.

Means and Their Inequalities 255 Suppose that к + £ > η; then a\ and or- ' have m,m = m(i, j, k, £), elements in common, m t^O, 1 <i<j <X= ("). For convenience we introduce the following notations: Lemma 10 lfk + £>n then η =Я1®{тф 1 < г < к), 1<3<К (14) σ· =Wl[*\aij,l < j < λ), 1<г<к. (15) D We only give a proof of (14) as that for (15) is similar. Further if t = =too the result is immediate and so we will assume that t is finite. Let us also assume that t ^ 0. Under these assumptions the sum on the right-hand side of (14) is a linear com- (£) bination of t-th powers of the elements of a- . Since the summation extends over all the /c-tuples a\ the sum is unchanged if the elements of щ are permuted. Hence the right-hand side is a multiple of ту, and by taking a constant it is seen to be actually equal to Tj. The case t = 0. can be handled in the same way. D Theorem 11 If -co < s <t < oo and k + £> η then mn{t,s;k;a) < OTn(s, t; £· a), (16) with equality if and only if α is constant. D We only consider the case where s, t are finite and non-zero as the other cases are either trivial or similar. Using (11), Lemma 10, 3.1.3(9) and (r;s) the following can be established. Tln{t,s-k]a) = m[^](aul <i < к) =Ш[^] (Τϊψ(σί3,1 < j < λ), 1 < г < κ) <τιψ (mfiny ι < j < λ), ι < г < κ) =OT^(rj, 1 < j < λ) = mn{s, t- £;a). This gives (16) and the case of equality follows from that of 3.1.3(9) and (r;s). D Remark (ii) Taking k = £ = nwe see, from Example (i), that both (12) and (16) include (r;s) as a special case.

256 Chapter III Consider the following 2 χ η matrix 9Jln(s,i;l;a) Wln(s,t;2;a) ... Wln(s, t; n-\ 1; a) 93ΐη(Μ;^;α) OTn(i,s;n;a) 0Jtn(i, s; η - 1; a) ... OTn(i, s; 2; a) OTn(i, s; 1; a) where we assume that s < t and α is not constant. The inequalities (12) and (16) can be summarized as saying: (i) the rows of Μ are strictly increasing to the right; (ii) the columns, except the first and the last, strictly increase downwards; and of course the entries in the first column both equal SDT^(a), while those in the last column both equal 97t™(a). Example (ii) Taking η = 3, t = 1, s = 0, к = 2, a = (а, Ь, c) in (16) gives the inequality Vab + Vbc+ л/са Ma + 6) (6 + c)(c + a) f 3 -V I ; (17) a further discussion of this inequality, that can be compared with V 5.4(35), is in [Carlson 1970a]. Example (iii) Another particular case of (16) is given when t = 1, s = 0, к = η — 1, *η(0η-ι(αί); 1<k<n)< ΜΟη-ιίαί); 1 < ^ < ^); see [ΑΙρ.379], [Carlson 1970b]. Remark (iii) Further results can be found in II 3.5 Remark(iv), V 5.4 and other mixed means are discussed in II 3.4, and 3.2.6 above; see also 6.2 Remark (i) and [Acu; Mitrinovic Sz Pecaric 1988b; Ozeki 1973]. 6 Some Other Results In this section are a few results on power means that do not fit into the above discussion. Still more results can be found in the references: [Jecklin 1963]. 6.1 Means on the Move Hoehn and Niven proved a very interesting result, called means on the move, that was generalized by Brenner and others; [Aczel L· Pales; Brenner 1985; Bullen 1990; Hoehn Sz Niven]. If a, w_ are η-tuples, and r £ Ш then the result of Hoehn and Niven considers the behaviour of ^щ} (α 4- te\ w) as t —► oo. Theorem 1 With the above notation, lim (#4r] (a + te; w) - 2in(a + te; w)) = 0; (1) t—>oo '

Means and Their Inequalities 257 equivalently, lim (ЯЯМ (α + te; w) - t) = 2in (а; ш) ■ D Let us first note that both of these limits exist. If we put f(t) = UJv£ (a + te; w) — 2ln(a + ie; υ;) then a simple calculation and 1(2) give mina — 2ln(a; w) < f(t) < max a — 21п(&;ш)5 so / is bounded. Simple calculations show that if Mr (i) = QJtJ^ (a 4- ie; w) — 2ln (a + ie; υ;) then Μ/ = (Mr_i/Mr)r~ and so /' = (Mr_i/Mr) - 1. Using this we have by (r;s) that /' < 0. So / is a bounded decreasing function, lini£_+oo f(t) exists and is in the interval [min а — 2ln (α; υ;), max а — 2ln (α; υ;)]. If r ^ 0 then η ΜΓ(ί)-Μι(ί)= (i(^-1^^(i + ai/i)r)l/r-l-2in(a;^)) = 0(1/ί), ί -> oo. by Taylor's theorem8. This completes the proof in this case. If r = 0 then by (GA), 0 > MQ(t) - M^t) > M_i(i) - Mi(i), so the result follows from the r^O case. D Remark (i) It follows from the above argument that if r < 1 < s and if α is not constant then M'T > M's. This is not in general true if r, s do not straddle 1; see [Hoehn L· Niven]. Remark (ii) Generalizations of this result can be found in IV 4.5 and VI 4.3. Another interpretation of this result is that translating the data values to larger values pushes the power mean of the translated values closer to their arithmetic mean. Alternatively the power mean of the translated values translated back moves closer to the arithmetic mean of the original values; [Farnsworth Sz Orr 1988; Wang С L 1989b]. The idea in the last remark has been used to introduce a further generalization of the power means; [Persson L· Sjostrand]. Definition 2 If α and w_ be η-tuples with Wn = 1, and t > 0 define f ((ΣΓ=ι«'ί(β?+*),,/,)*/ρ-*)1/ί> if^°- Ι (Σ?=1 Ща?)1'*, if ρ ^ 0, <? = 0. See I 2.2 Footnote 2.

258 Chapter III Theorem 3 With the above notations, and q^O (а) т^(ат) = Ш^°(а-шУ, t—ЮО (с) 9Я[?'91;в(а;ш) < ОТ{?'*]*(а;ш), s < *. D (a) This is immediate. (b)(t)p^o ^ i=i ' =ί1/'((Σω·(1 + ^+°(Γ2))),/,-1)1/' by I 2.1(5), =*!/*((! + H^Bfi + 0(i-2))?/P - l)1/? =i1/?((l/i)(f>af) f +0(r2))1/g, by I 2.1(5 ), i=i =0n^](a-w)+O(t~1), t -f oo. (w) ρ = О ше ^(а;иО=^(Ц(1 + ^Г-1) 1/9 =ti/,(l + ΣΓ^ια? + 0(r2) _ ^ by j 2Λ(5)ϊ =^1?](а;ш) +0(t~1), t -> oo. (c) We will only consider the case 0 < q < ρ as the other cases are similar. The proof of Theorem 1 shows that if s < t then m%/q]{a* + s;w) < m%,q\aq + t)w). Raising both sides of this inequality to the power \jq gives the result. D 6.2 Hlawka-type Inequalities One theme in this book has been the study of sub-additivity or super-additivity of the difference between two sides of various inequalities, the difference being regarded as a function on the index set; see I 4.2 Theorem 15, II 3.2.2 Theorem 7, above 2.5.2 Theorem 12, 3.2.4 Corollary 18, and below IV 3.1 Theorem 1, Corollary 2. In an interesting paper Bur kill, [Burkill], considered replacing the super-additivity of a function σ on the index sets by an

Means and Their Inequalities 259 inequality related to convexity, called H-positivity] [PPT p. 176-178]. Let I, J and К be disjoint index sets the set function σ is Η-positive if: σ(Ι U J U К) + σ(Ι) + a{J) + σ(Κ) > σ{1 UJ)-f σ(J U К) + σ{Κ U /); (2) An inequality of type (2) was called by Burkill a Hlawka inequality, because of the basic result for triples of η-tuples due to Hlawka: |a + 6 + c| + |a| + |£| + |c| > \a + 6| + \b + c\ + |c + a| : see [DI pp.125-127]. The general result for η-tuples ai5 1 < г < га, Σΐ^ίΙ- Σ lfii + ail+ Σ la< + 0J+a*l + (-i)m_1l«i + ---aml>o ϊ = 1 l<i<j'<m l<i<j<fc<ra holds in general only if m = 1, trivial, m = 2, (M), or ra = 3, Hlawka's inequality; [Jiang Sz Cheng]. In his paper Burkill proved several results of this type by considering : (a) σ(Ι) = W/tf/few), and (b) σ(Ι) = И^(Ш#*](а; w)), where φ is a function with φ" - (r - 1)0' > 0 ; in all cases J С {1, 2,3}. In particular he obtained the following result. Theorem 4 if α and w are 3-tuples, with W3 = 1 then: 0з(а;ш) + Яз(а;ш) > (ил + w2)ar/(wi+waWa/(W1+Wa) + (w2 + ^3)аГ/("1+"2)% 3/("1+"2) + (ws + ^ι)α^3/("1+"2)αΓι/(ωι+"2). If in addition r > Q and φ :R+^ R, with φ" - (r - 1)0' > 0 then: > Οι +ы2)ф(Ш[2](а1,а2;'ш1,'Ш2)) (3) + (ΐϋ2 + ^з)0(Ш12г](а2,а3;г(;2,^з)) 4- (w3 + wi)0(OT2r](a3, αλ\ u>3, ил)). The methods of proof were elementary and the paper caused much interest. In particular if in the second case r = 1, when the mean is the arithmetic mean, the condition on φ implies that it is convex. This simpler condition has been used to obtain the same result, see [Baston 1976; Vasic &; Stankovic 1976], and [Pecaric 1986] for a different proof. It was Pecaric who noted that (3), with r = 1, implies the following more general result; [PPT pp.171-180].

260 Chapter III THEOREM 5 If φ : [α, b] ι—► R is convex, υ; ал η-tuple, a a real η-tuple with α < си < b, 1 < г < η, and if 2 < к < η then, Wn(^~^^ (4) Inequality (3) with r = 1 is the case η = 3,/c = 2 of (4); further in the case φ{χ) = \x\ (4) has been called Djokovic's inequality and is equivalent to the Hlawka inequality; [Djokovic 1963; Takahasi, Takahashi L· Honda; Takahasi, Takahashi L· Wada]. Remark (i) The above results imply certain inequalities for mixed means due to Kober, II 3.5 Remark (iv), [Kober; Mesihovic; Ozeki 1973). 6.3 p-Mean Convexity The main results of this chapter can be, and have been, derived from the properties of convex functions. It is possible to reverse this process. If — oo < ρ < oo, a, b > 0 define mp(a,b)=i^lpi(a,b) if ab φ 0, Ιθ if either a = 0 or b = 0. We then say that / : Rn н-► [0, oo[ is p-теап convex if: f(Xx + (1 - X)y) <mxp{f(x), f(y)), for all x,y, and λ, 0 < λ < 1. There are obvious definitions of p-mean concavity, p-mean log-convexity etc.; [Das Gupta; Uhrin 1975,1982,1984]. In particular Uhrin has proved the following theorem. Theorem 6 If ρ -f q > 0, if f is p-mean concave, and д is q-mean concave then the convolution f * g, f * g(u) = J^f(u — n)p(v)dv, is (p_1 -f- q~x + r)~1-mean concave if— < < oo. r p + q Following the comments in II 1.2 Remark (ii) we can make a completely different definition and say that a positive function is p-th mean convex, pet, on [a, b]9 if for all x,y,a < x,y <b and λ, 0 < λ < 1, f(\x + T^Xy) < m^{f(x),f(y);\, 1 - λ)); equivalently if ρ φ 0 this is just the statement that fp is convex; if the inequality is reversed we say that / is p-th mean concave on [a, b] 6.4 Various Results Here the terminology p-convex is more usual but when ρ is an integer this is confusing; see I 4.7.

Means and Their Inequalities 261 Theorem 7 (a) If α is an η-tuple and the η-tuple b is denned by b\ = αϊ, bi (di 4- Oi_i)/2, 2 < i < n, then, with the notation of 4.1, Q^(6)<2Q^(a). (b) If η > 2 and α is an η-tuple of distinct entries then where ^p,n — 2^-1 Elnj;1)/2(2^, if η is odd, min{2p -1,1} Ση=ι№ - !)p> if ™ is even. (c) Let f : [0, a] ^ Μ be (fc + l)-convex, with /(w)(°) = °> 1 < m < fc - 1, a a real η-tuple with entries in [0, α], υ; an η-tuple then f{Tt\a;w))<Vin{f(a);w). (d) Let a be an m-tuple, г ЕШ, and denne the sequence recursively by ап = 9ЯЙ (a<n-m, · · ·, αη-ι), n>m. Then lim an = <( V(w2+1), mi an = < V ( 2 ) / { {<бш{а\,1 < t < m))w/(W2+1), if r = 0. n—>oo D (a) See [Math. Lapok, Problem F 1930, 50 (1975), 11-12]. (b) The case η = 2 is in [Mitrinovic, Newman L· Lehmer]; see also [AI p.34-0], The case ρ > 1 was stated without proof in [Ozeiri 1968]] a proof was given in [Mitrinovic L· Kalajdzic]. A proof for ρ < 1 is given in [jRusseii]. (c) This is a generalization, by Vasic, of a result of Markovic. A different proof has been given by Pecaric who also shows that the same inequality holds if f{x1^k) is convex on [0, ak]; [Keckic L· Lackovic; Markovic; Pecaric 1980a; Vasic 1968]. (d) See [Elem. Math. 29 (1974), 15-16]. D Theorem 8 If η > 2 and w is an η-tuple, Wn — 1, α a reai η-tuple with m < α < Μ, and ifr<s,t^ 0, then 1 |r(r-t))| < (iWLrl(a;m))r-(^]few))r < 1 |r(r-t))| M(.-r) | e(r _ i) | - (2RW(a;«;))s - (»# (α; ω))' " m<-r) I s(r - i) I

262 Chapter III Remark (i) This result is due to Mercer and the proof uses his mean-value theorem, II 4.1 Lemma 3. If s = 2, r = 1 and t —> 0 the above inequalities reduce to those of II 4.1 Theorem 2(b); [Mercer A 1999). Compare this result with the first part of 3.1.1 Theorem 2. The following is an extension of Sierpinski's inequality, II 3.8 (57); [Alzer, Ando Sz Nakamura]. Theorem 9 Иа,ш are a decreasing η-tuples, η > 3, then ifr > 0, (^1fem))w"-1(^r]feM))i"" with equality if and only if α is constant. D This follows from II 3.8 Theorem 27 by taking χ — 0 and χ = r. D Remark (ii) II 3.7 Corollary 28 is the case t = 1 of this theorem. As in that case we note that it is sufficient to assume that the η-tuples α, w be similarly ordered, see II 3.8 Remark (viii). Remark (iii) Further interesting properties of the left-hand side of (5) can be found in [Pecaric 1988; Pecaric L· Rasa 1993). In the case of the special sequence y_ — {1,2,...} certain special results are obtainable of which the most interesting is perhaps the following. Theorem 10 If r > 0 and n>\ then " <Jf<£L< ^^. („ D The left-hand inequality is obtained by using the function f(x) = xr in I 4.1 Lemma 8. The same lemma with f(x) = log ж gives the outer inequality; it remains to prove the right-hand inequality. The right-hand inequality is obtained if we show that vn — (Σ™=1 ir)/n(n!)r/n, η = 1, 2,... is an increasing sequence. Now define the two decreasing n{n — 1)- tuples, by: (n — ]A for 0 < к < η - 1, afc(n_!)+1 = a/c(n-i)+2 = ■ ■ ■ = a^n-^+n-i = , m/n ' and for 0 < к < η - 2, bfcn+i = bfcn+2 = ■ · ■ = bkn+n = г ((n-1)) Then vn > vn-i is equivalent to n(n—ι) η(η—ι) l/(n-l)·

Means and Their Inequalities 263 To see this it suffices, from I 4.1 Corollary 11, to prove that га га Y\bi < ТТсц, 1 < m < n{n — 1). г—ι г—ι Now when m = n(n — 1) both sides of this last inequality are equal, and equal to 1. For some unique /c, j, 0 < к < η — 1, 1 < j < η — 1, m = /c(n — 1) + j and then Πα ("Ο"'1 ("~*0' Π г ((n _ Л _ ι),)""1 (n![fc(n-i)+4/n) · Note that m = k(n — 1) 4- j = kn 4- (j — /c); so there are two possibilities: (i) j — к > 0. (ii) j — /c < 0, when we write m = (k — l)n + n -{- j — k. Evaluating Πΐϋι ^ m these two cases and comparing with the above value of Πΐϋι аг gives the result; for details see either the paper of Martins or that of Alzer. D Remark (iv) The outer inequality in (6) is called the Minc-Sathre inequality, [DIp.85], [Mine &Sathre); the right-hand inequality is due to Martins, [Martins]. The final form above is due to Alzer, [Alzer 1993d; Chan, Gao L· Qi; Kuang; Qi 1999a, 2000a; Qi L· Debnath 2000a; Sandor 1995b]. Remark (v) Considered as bounds on the central ratio the extreme terms are best possible as was pointed out by Alzer. Remark (vi) Martins has pointed out that in the case of a^ = гг, г > 1, his inequality improves the equal weight case of (P), being just »η+ι(α) > Яп(й) 0n+i(u) 0η(α)" This lead Alzer to further observe that the following improvement of (R) holds in this special case: 2ln+i(a) ~ #η+ι(α) > Мй) ~ ^n(a)· The following result concerning the factorial function due to Alzer; [Alzer 2000a]. Theorem 11 There are two positive numbers α, β, 0.01 < a < β < 11.3, such that inequality (artM(a;te))!<anW(a!;te)

264 Chapter III holds for all positive η-tuples a, u>, with Wn = 1, if and only if а < r < β. Whenever the inequality holds it is strict unless α in constant. Remark (vii) For the exact definition of the two constants α, β the reader should consult the reference. Remark (viii) The case r = 1 is just (J) applied to the strictly convex factorial function; I 4.1 Example(ii). Remark (ix) In the case of harmonic and geometric means, when r is outside the above interval [α, β], it is known that the inequality only holds for certain n-tuples a; see the above reference. A related factorial function inequality, also by Alzer, says that й((х!)2,(^)2)>1. This generalizes a well-known inequality of Gautschi where the harmonic mean is replaced by the geometric mean, see [DI p.84], [Alzer 1997c, 2000b]. The following result is due to Mercer, [Mercer A 2002], generalizes II 5.9 (15). Theorem 12 If a and w are η tuples with α not constant, m < α < Μ, and Wn = 1 then, denning M^ = M® (M, m; a) = (m* + M* - (Wl[*](a; w))*) , t e R, ifr,seR withr < s, m < M[r]{M, M[s](M,m;a) <M D Using the substitutions in 1(3) and 1(4) it suffices to consider the inequality m;a), t > 1. Further we can, without loss in generality, assume that a is increasing. Let V = MM(M,m;a) - AfW(M, ra;a) then simple calculations show that: dV/doi = ^(1-(^/М^)*_1),апаа|М^-а,|/ааг· = ±(ηΗαΙ~1(Μ^ψ-^/4ΐ), according as aL > М^,а{ < M[il Using these relations we can let the a-i < M^ tend to m and those a.L > Μ Μ tend to Μ, to get for some positive W\, W2,W\ -f W<i = 1, V > ((1 - Wi)mb + (1 - \¥2)Мг)1/г - ((1 - Wi)m + (1 - W2)M)\ and so V > 0 by (r;s). D Remark (x) II 5.9 (15) is just ΜH (M, m;a) < M^ (M, m;a) < m\ a).

Means and Their Inequalities 265 Remark (xi) It is readily checked that limi_+_00 M^(M, m; a) = m, and that Km М^(М,т; a) = Μ. Finally we remark that if r G Μ then Wl%> (a), has the property of ra-associativity, 1 < m < n, see II 1.1. It has been proved that if two means, one defined on n-tuples and the other on rrt-tuples means, have this property and if both homogeneous and symmetric and one at least is an analytic function then the means are r-th power power means for some τ G M; [Kuczma 1993).

IV QUASI-ARITHMETIC MEANS The power means are defined using the convex, or concave, power, logarithmic and exponential functions In this chapter means are defined using arbitrary convex and concave functions by a natural extension of the classical definitions and analogues of the basic results of the earlier chapters are investigated. First however we take up the problem of different convex functions defining the same means; the case of equivalent means. The generalizations (GA) and (r;s), their converses and the Rado-Popoviciu type extensions are studied under the topic of comparable means. The definition can be further extended although this leads to the topics of functional equations and functional inequalities so is not followed in detail. 1 Definitions and Basic Properties 1.1 The Definition and Examples The power means ^{a-w), r GR, defined in the previous chapter can be looked at in the following way. If r £ R and χ > 0 define rhi \-iX^ if Г^0' nX)~\\ogx, ifr = 0; then Ш1\^ш) = ф-4^-^1ф{а,)). (1) This suggests the following definition; [DI pp.217- 218; HLP pp.65-70]. Definition 1 Let [m, M) be a closed interval in R, and Μ : [га, M] ь-+ R be a continuous, strictly monotonic function; let α be a real η-tuple with m < α < Μ, ui be a non-negative η-tuple with Wn φ 0; then the quasi-arithmetic Л4-теап of α with weight w. is OTn(a; w) = Μ'1 f-L ]Г илМ(аА . (2) The function Μ is said to generate, to be a generator or to be a generating function of the mean fXfln. 266

Means and Their Inequalities 267 In addition we will sometimes speak of the quasi-arithmetic mean, or just the Л4-теап, of α with weight w_. In the case of η = 2 the above definition has a simple geometric interpretation. У=М(х) M(a2) w M^) Figure 1 9Jl(av& yWl.w2) Remark (i) The functions used to define the quasi-arithmetic means will be written Λ4, ΛΛ Τ etc1, and the corresponding means will then be 9Я, 9T, $ etc. Letters such as 21, 0,й,П that already denote particular means will be avoided if there is any possibility of ambiguity. These functions are finite except perhaps at the end-points of their domains. Remark (ii) The function M~x is continuous, strictly monotonic and defined on the closed interval min { Л4 (ra), Λ4 (M)}, max bince we can write Tln(a;w) = Μ'1 Ып{М(а)',уи)} the internality of the arithmetic mean, II 1.2 Theorem 6, implies that (Σ™=1 WiM{ai)/Wn) is in the domain of Л4-1, and so the right-hand side of (2) is meaningful. Remark (iii) As usual Шп(а) will indicate the mean with equal weights, and a, and, or w_, may be omitted when there is no ambiguity; further 9Я/(а; w) will have the meaning expected when i" is an index set; see Notations 6(xi), II 3.2.2, III 2.5.2. Remark (iv) As will be seen below, 1.2 Remark (ii), there is no loss in generality in assuming that M. is strictly increasing, although in particular examples this is not always the case. Remark (v) Note that in this chapter we do not assume that the η-tuple a is Except in Figure 1 where Μ replaces Ad.

268 Chapter IV positive2, just that m < a< M; a is always finite even if either ra, or M, is not . Remark (vi) We also allow w to have zero elements. In general this allows the statement of inequalities to include the case of all k, 1 < к < η, in the statement of the к — η; then case of general к arises when w_ has η — к zero elements. So there is a need to restate the cases of equality as the conditions need only apply to those elements of a with non-zero weights, that is to the essential elements of a; see I 4.3, II 3.7. In the following discussions all the η-tuples a,w_, and the functions will be assumed to satisfy the various conditions in Definition 1. However in applications to particular cases and in the discussion of Rado-Popoviciu type results we will for simplicity take w to be positive. In addition in the case where several means are involved it is clear that the η-tuples a must be restricted to the intersection of the various intervals [m, M] involved. The following lemma states some simple properties of quasi-arithmetic means. LEMMA 2 (a) The quasi-arithmetic M- mean has the properties: (Sy*), (Sy) in the essentially equal weight case, essentially internal, essentially renexive, (Co) and, if ЛЛ is increasing, (Mo). 1 n (b) There is a unique a, min a < а < max a such that Л4(a) = -—- (\J WiM(ai)). Furthermore unless a is essentially constant, some αι is less than a, ana some αι is greater than a. (c) If η > 2, a a fixed n-tuple and α is given with min a < а < max a, then there is a non-negative η-tuple w_ with Wn = 1 such that 9Rn(a;w_) = a; further this w_ is unique if and only if η = 2. (d) Ifu, v_ are non-negative η-tuples with Un =£ 0, Vn ^ 0 and ifu/Un < v_/Vn then Шп(а;и) <Жп(а-у). D All of (a) is trivial; the terminology in (a) is explained in II 3.7. (b) This is immediate from Definition 1 and by noting that Y^=1 Wi{M.{a) — M(ai)) = 0, which implies that unless all the terms of this expression are zero some must be positive and some negative. (c) This is an immediate consequence of the fact that, given a, Wln(a;w) is а continuous function from the compact set of υ; to the closed interval [min a, max a]. (d) This is trivial. D That is, Convention 3 of II 1 1 will not apply in this chapter.

Means and Their Inequalities 269 Remark (vii) This lemma justifies the use of the word mean and generalizes parts of II 1.1 Theorem 2, II 1.2 Theorem 6 and III 1 Theorem 2. Remark (viii) These means are discussed in detail in [HLP pp.65-101), [Mitri- novic & Vasic pp. 104-113], [Aczel 1948а,1956а,1961а; Chisini 1929; de Finetti 1930,1931; Jecklm 1949a,b]. From the remarks preceding Definition 1 the power means form an example of a scale of quasi-arithmetic means. Further examples of quasi-arithmetic means, and scales of quasi-arithmetic means, are given in the following examples. Example (i) [Burrows L· Talbot; Rennie 1991] If 7 > 0 and if [m, M] = 1 let 4 ' [x, if 7 = 1. Then 9Лп(а;ш), written in this case OT7,nfe ш), is ^7,nfe^L) = { ^21п(а;ш), if 7=1. Properties of these means are given below; see 2 Example(vi), 5 Example (iv). Example (ii) If 7 > 0, 7 =£ 1, taking M{x) = y1/* gives another family of means, called the radical means: Шп(а;Ш) = (ΐο^Ι^^^Ί Example (iii) If M{x) = xx, χ > e 1, we get the so-called basis-exponential mean; if its value is μ then Example "(iv) If Λ4(χ) = χγΙχ, χ > e"1, then in a similar manner we get the basis-radical mean; if its value is ν then ν 1 1^ = —^Гт,п^а· Wn< Remark (ix) The above means are some of the many introduced and used by the Italian school of statisticians; see [Bonferroni 1923-4,1924-5,1927; Gini 1926; Pizzetti 1939; Ricci]. The following results can be obtained.

270 Chapter IV Theorem 3 If a is a positive η-tuple then Σ(*η(α)Γ<±α?; ±a^ <±^n(a))^; 3*1^ (a) < I £>?«; i=i i—i i—i i=i i=i η η ^(OT[[](a))ai < ^ a», where r < тах{а^; а» < ш4г](а)}. with equality if and only if α is constant. D See [Pizetti 1939]. D Example (v) Various trigonometric means have been studied; see in particular [Bonferroni 1927; Jecklin 1953; Pratelli]. If 0 < а < π/2 Θη (α; υ;) = arcsin ί ^~ Σ™=1 гиг- sin а^), £n (а; ш) = arccos ί ^ Σ"=1 Щ cos α» J; if 0 < а < π/2, Χη (α; υ;) = arctan ί -ψ- Σ7=ι Wi ^an α*)> and if 0 < а < π/2, <£Χη (α; w) = arccot ί ^- Y^i=± wi co^ a*) ■ The following result is due to Jecklin; [Jecklin 1953}; see 5 Example(xi); see also 2 Example(iii). No proof will be given as the theorem follows from more general results given later. Theorem 4 Let α be a positive non-constant η-tuple and write α, β, 7, δ for the solutions in [0, π/2] of tan χ = 1/л/2, cot χ = 2x, cot χ = χ, tan ж = 2x, respectively and e, φ for the solutions in ~Щ_ of cot χ = 1/л/2, tan ж = 2/χ respectively. Then: S)n(a) < €%η(α) < ®n(a) < 6η(α) < Χη(α) < ^(α) < Πη(α), 0 < α < α; Йп(а) < £Γη(α) < 0η(α) < 6η(α) < £η(α) < Χη(α) < Πη(α), &<α<β; Йп(а) < ^η(α) < 0η(α) < ©η(α) < Μ«) < Οη(α) < £η(α), /? < α < 75 Йп(а) < ^Γη(α) < 6η(α) < 0η(α) < £η(α) < 0η(α) < Хп(й), 7 < α < е; Йп(й) < ©η (й) < б^п(а) < 0η(α) < £η(α) < On (α) < £п(й), 7 < й < е; ©η(а) < #η(α) < ^η(α) < βη(α) < €η(α) < 0η(α) < Χη(α), 0 < α < <*; ©η (α) < Йп(й) < ^η(α) < £Γη(α) < £η(α) < Οη(α) < ^η(α), <* < α < π/2· Remark (χ) A somewhat related mean is the following defined in [Ginalska]. Let а > 1 then the exponential mean of α is: <£п(а) = ехр((ПГ=1loga^)1/71). While it is not difficult to show that <£n (a) < 0n (a), the exponential and harmonic means are not comparable.

Means and Their Inequalities 271 1.2 Equivalent Quasi-arithmetic Means It is natural to ask how far the knowledge of the quasi-arithmetic .M-means determines the function Λ4; or, when do two functions generate the same mean? Equivalently: if-for two functions Μ and TV the quasi-arithmetic Л4-теап of a with weight w_ is always equal to the quasi-arithmetic ЛЛ-теап of a with weight w, is Μ = ΝΊ This is answered by the following result due to Knopp and Jessen; see [HLP pp.66-68], [Jessen 1931b; Knopp 1929}] a more general result can be found in [Bajraktarevic 1958). THEOREM 5 in order that Tln(a;w) = У1п(а;уи_), for all η and all appropriate η-tuples α in the common domain of Μ and λί and all non-negative w, Wn -φ 0, it is necessary and sufficient that Λ4 = αλί -f β for some a,^Gl,a/0. D Suppose first that the condition holds, then: 1 n M(mn{a;w)) =--i^wiM{ai)) i=i =od\f(mn(a]w)) +β = M(Vln(a;w)). So, by 1.1 Lemma 2(b), 9Лп(й;ш) = 9Тп(й;ш)· Now suppose that Ш2{а\п1) = У12(д/,и1) for all appropriate 2-tuples a and w_, assuming, without loss in generality, that w\ -f w2 = 1. That is Μ"1 (ΐϋιΛ^ί(αι) + w2M{a2)) = λί"1 (υ^Λ/Χαι) + w2N{a2)). Putting φ = Μ οΛ/""-1, Xi = N{ai), г = 1,2, this is equivalent to saying that for all (zi,z2), </)(wiXi 4- ι^2χ2) = ίυ1φ(χ1) 4- ι^2φ(χ2). Hence by a well known result, see [Aczel 1966 p.49], ф{х) — αχ Λ- β for some α,β eR, a^O. Thus ΜοΜ"λ{χ) = ax 4- β, or Μ = αλί 4- β. □ Remark (i) This actually proves more: Μ = αλί4- β once 9ft2(a',w) = Ul2(a; w) for all appropriate 2-tuples a and w_. Remark (ii) An immediate consequence of this result is that Μ and — M. define the same mean. Hence in 1.1 Definition 1 we can, without loss in generality, require that Λ4 be strictly increasing.

272 Chapter IV Example (i) Using this result it is possible to fit the geometric mean into the family of power means in a neat manner. If r e R, define J^{x) = J^ tr~~1 dt; that is ( xr -I r , r ι > ifr^O I log ж, if r = 0. Then for г = 0 the quasi-arithmetic .F^-mean is the geometric mean, and by Theorem 5, for all other values of r it is just the r-th power mean. A similar procedure can be used to fit the mean 9Jtijn into the 9Jt7jn family of means; see 1.1 Example (i). As a result of Theorem 5 it is possible to define a metric on the class of all continuous strictly increasing functions ЛЛ on [m,M] normalized by Ai(m) = т,Л4(М) = Μ. If Λ4 and λί are two such normalized functions let η e N* be fixed and define ρ(Μ,λί) = sup{£; t = |9Яп(а;ш) -*Яп(а;ш)|, Ш> 0,m < a < Μ}. This metric defines a metric space that is homeomorphic to the one using the sup norm; it is bounded but not totally bounded; see [Cargo L· Shisha 1969]. As we saw above, the power means are the basic quasi-arithmetic means; we now show that they are the only homogeneous quasi-arithmetic means. Theorem 6 Let M. :]0, oo[i—► Ш be continuous and strictly monotonic and suppose that for all к > 0, and all positive η-tuples α and w_ DJln (fca; w) = кЖп (a; w). (3) Then for some rGl and all positive η-tuples a and w_, Wln(a;w_) = Ш^/(а]ш)· D Clearly the power means satisfy (3). By Theorem 5 we can assume that M{1) = 0. By (3) 9#n fe W.) — к ~1 Wln (ka; w) = &n (a; w), where JC(x) = M(kx). By Theorem 5 there are a(fc), p(k) e R, a(k) φ 0, such that К = a(k)M + p(k). Further since M(l) = 0 we have that p(k) = /C(l) = M(k). Hence for all positive x, у M{xy) = a(y)M(x) + M(y). (4)

Means and Their Inequalities 273 Oi[X) — 1 Οί\ΊΙ ι — 1 Interchanging x,y and subtracting gives t ;, . = ; * ч , or (a — 1)1 Μ is a Мух) М(у) constant, с say. Substituting this in (4) gives the following functional equation for M: M{xy) = cM(x)M(y) + M(x) + M(y). (5) There are two cases to consider: c = 0,c^0. Case(i) c = Q Then (5) reduces to M(xy) = M(x) + M(y), (6) of which the most general continuous solution is M(x) = A log χ; see [Aczel 1966 p.48]. So, by Theorem 5 3Κη(α;υ;) = 0п(а;;ш). Case(ii) c^O Putting λί = 1 + cM, (5) is equivalent to Af(xy) = Af(x)Af(y), of which the most general continuous solution is Af(x) = xr\ see [Aczel 1966 p. 48)- So M(x) = (xr - l)/с and by Theorem 5 UJln(q-w) = Wltf (a; w). D Remark (iii) The above proof is a simplification of one due to Jessen, and can be found in [HLPpp.68-69); [Ballantine; de Finetti 1930,1931; Jessen 1931a; Nagumo 1930; Poveda). 2 Comparable Means and Functions We now turn our attention to generalizations of (r;s) that are valid for quasi- arithmetic means. The result, Theorem 5 below, provides yet another proof of (GA), as well as of (r;s); it seems to have first been proved by Jessen; [HLP pp.69-70,75-76), [Dinghas 1968; Jecklin 1949a; Jessen 1931b). Definition 1 The quasi-arithmetic M-mean and the quasi-arithmetic λί-mean are said to be comparable when: (a) the functions Μ and λί have the same domain; (b) for all η and all appropriate η-tuples α and non-negative w, Wn =£ 0, either mn{a;,w) <9Tn(a;w), (1) or for all η and all appropriate η-tuples α and non-negative w, Wn =£ 0, (~1) holds. Remark (i) Thus (GA) says that the arithmetic and geometric means are comparable, and (r;s) says that any two power means are comparable.

274 Chapter IV Lemma 2 If Μ and N are continuous functions on [ra,M], and if N is strictly monotonic and convex with respect to Μ then М{Жп(а;щ)) <Кп(Я(а)-щ) (2) for all η-tuples α such that m < α < Μ, and all non-negative η-tuples w. with Wn t^ 0. If N is strictly convex with respect to Μ there is equality in (2) if and only if α is essentially constant. If N is concave with respect to Μ (~2) holds. D Putting b = Μ(α), φ = Ν ο M~l then φ is convex, see I 4.5.3, and (2) reduces to 0(21η(έ; υ;)) < %п(Ф(Ь)',ш), which is just (J), I 4.2 Theorem 12. The rest of the theorem follows from the cases of equality in (J), and from the fact that if φ is concave then —φ is convex. D Remark (ii) In particular cases some care is needed in choosing the domain [m, M]. The function λί ο М~г has to be convex on [Л4(га), M(M)], so assuming that Μ is increasing we must choose m and Μ so that M(m) and Λ4(Μ) lie in the domain of convexity of Ν ο Mr1. Remark (iii) This simple result has been rediscovered many times; see [Bonfer- roni 1927; Bullen 1970b,1971b; Cargo 1965; Chimenti; Jecklin 1947; Lob; Shisha L· Cargo; Watanabe). The hypotheses Lemma 2 are weaker than those of Theorem 5 below so this result can be used to deduce inequalities between means not obtainable from that theorem. Corollary 3 If 0 < r < s < oo, α an η-tuple with а > ((s -r)/(s + r))r , w_ a positive η-tuple, then !*» < y- _^_ f3) 1 + (^(«;м))5-^1 + «Г У> with equality if and only if α is constant. D If r t^ 0takeM(x) = xr, N(x) = 1/(1 + xs) thenA/O^M-1 is strictly convex provided 0 < r < s and xslr > (s — r)/(s + r). The result is then immediate from Lemma 2. If r = 0 the same argument applies taking M(x) = log ж. D Example (i) When s = 1 and r = 0 (3) becomes 1 + 6η(α;υ;) 4^ 1 + α%' N ' г=1 <> Τ^-,α>1; (4)

Means and Their Inequalities 275 and if a < 1 (~4) holds. The equal weight case of (4) is due to Henrici and so (4) is known as HenricVs inequality. In general inequality (4) cannot be improved by replacing the geometric mean by an r-th power mean, r < 0. However the following related result has been proved by Brenner & Alzer: Wi 1-сц S < — - l- > — , - ι- 1 r η -S)n(a]w) if a < 1, if a > 1; also in this case the inequalities cannot be improved by using other power means in the denominators . See [AIp.212; DIpp.121-122], [Brenner L· Alzer; Bullen 1969a; Kalajdzic 1973; Mitrinovic L· Vasic 1968c]. The proof of Corollary 3 depends on the strict convexity of the special function Λ/Ό M~l defined there. If then we consider any function / such that 1/(1 + /) is strictly convex on [m, M], (3) can be generalized as follows. Corollary 4 If the function f is such that 1/(1 + /) is strictly convex on [m, M], then Wn < A U)j ( 1 + /(Яп(а;щ)) -^1 + /Ы' ^ } provided m < α < Μ, and there is equality if and only if α is constant. Example (ii) Both Bonferroni and Giaccardi have used the strictly convex function ЛЛ(х) = ж log ж, see I 4.1 Example (i), to deduce from Lemma 2 that 1 n 2ln (а; Ш) bg 2in (a; w) < — ^ w^i log a»; Wn [Bonferroni 1927; Giaccardi 1955]. Example (iii) Using a definition in 1.1 Example (v) Lemma 2 can be used to prove that %п(а',Ш) < 6nfel), provided that 0 < α < π/2; [Lob]. We now give the main result of this section.

276 Chapter IV Theorem 5 Let Μ and N he strictly monotonic functions on [m, Μ], Ν strictly increasing, [respectively, decreasing], then for all η and all η-tuples a, m < a< M, and all non-negative η-tuples ui, Wn φ 0, ^nfew) <У1п(а;Ш) (6) if and only if Μ is convex, [respectively, concave], with respect to Λ4 If N is strictly convex,[respectively, concave], with respect to Μ there is equality in (6) if and only if α is essentially constant. If N is decreasing, [respectively, increasing], and N is convex, [respectively, concave], with respect to M. then (~6) holds. D This is an immediate consequence of Lemma 2. D Remark (iv) The cases of equality in Theorem 5, as well as conditions for (~6), can be obtained by a similar discussion. COROLLARY 6 The quasi-arithmetic ЛЛ-mean and quasi-arithmetic N-mean, with Μ and J\f having a common domain, are comparable if and only if either Μ is convex with respect to N, or N is convex with respect to M. Example (iv) In particular if M~l is convex then Шп{а\и)) < 21η,(α; w), while if M. is convex then UJln (a; w) > 2in (a; w). Example (v) If M(x) = xr, Af(x) = xs, rs =£ 0, then AfoM~1(x) = xslr', which is strictly convex if r < s, so (6) is a generalization of these cases of (r;s). The cases when either r = 0 or s = 0 can be discussed similarly by taking Μ or N to be log. Example (vi) Similarly, for the means in 1.1 Example (i), taking Λ4(χ) = μχ, Μ (χ) = νχ, μ, ν > 0, then Λ/Ό Λ4~1(χ) = ulog^ x which is strictly convex if ν > μ. So under this condition we have that with equality if and only if α is constant; this result is in [Bonferroni 1927; Pizzetti 1939]. Theorem 7 Let ΛΛ and N he strictly increasing functions on [m, M], and suppose that for all 2-tuples α,πι<α< Μ, and all non-negative 2-tuples w_, W2 ^ 0,

Means and Their Inequalities 277 and 9^2 few) < 9^2 few) then for all η, η > 2, and all η-tuples а,т<а< Μ, and all non-negative η-tuples w., Wn φ 0, inequality (6) holds. D Suppose that (6) has been established for all n, 2 < η < к. We can write 97ΐ*;_|_ι(α;υ;) = 9Я^(а;;ш), where di, ii 1 < i < к — 1, 9Jl2(afc,afc+i; ; > ; Ь lf г = ^ r Wi, if 1 < г < к — 1, w* = S The hypothesis of the theorem implies that α< α* where d;, if 1 < г < /c — 1, ^(afc^fc+r, ; , ; J, lb = /с; So by monotonicity, 1 Lemma 2(a), and the induction hypothesis, ЯЯ*+1(а;ш) = Wkfew) < Tth(a*;w) < Vth(a*',w) = 9^+1(а;;ш). D Corollary 8 Let ΛΛ,λί be strictly monotonic functions defined on [m,M] and let Μ. = αλΛ + β,λί = jAi + 5, where a, /?, 7, £ are real numbers chosen so that M{m) = ЛГ(т) = 0,M(M) = λί(Μ) = 1. Then inequality (6) holds if and only if the graph of M. lies above the graph of N; equivalently if and only if for all αι,α,2, m < αι,α,2 < M, and positive u>i, u>2 with w\ -f u>2 = 1, WlM(ai) + W2M(02) - M{wiai+W202) M(a2) - M(ai) wiAf(ai) + w2N'(a2) - Af(wiai -\-w2a2) < ЛГ{а2)-ЛГ(а1 D By 1.2 Theorem 5 the means in (6) are given by Μ,λί, as well as by Μ,λί; and by Theorem 7 it is sufficient to consider the case of η = 2. However in this case the result is immediate using the geometric interpretation in Figure 1. Π Other necessary and sufficient conditions for (6) have been given in [Mikusinski]. The first is that λί be less log-convex than Λ4 in the sense that for all distinct oi,a2 Δ2Λ4(αι,α2) Α2λίΜ(α1,α2) АМ{а1,а2) ~ ΔΛΓ(αι,α2) '

278 Chapter IV where: лллг,/ \ ka( \ ka( \ ь2кл( \ M(a1) + M(a2) Α,/αι+α2χ AMF{aua2) = M(a2)-M{a1), A2M(aua2) = —κ—^ —L-M{ ). Μ" Ν" The second condition is given in I 4.5.3 Theorem 34 : —— > ——; see also \Cargo Μ' Al' 1965]. A simple generalization of Theorem 5 is the following result. Theorem 9 If N is strictly increasing then /(а«п(а;ш)) <Яп(/(а);ш) if and only if No f о М~х is convex; further if this last function is strictly convex this inequality is strict unless α is essentially constant. In particular /(Яп(а;ш)) <М/(а);ш) if and only if f is log-convex. D The first part is immediate by applying (J) to the η-tuple Л4(а) using the function N о / о М~г. The second part then follows on taking Uln = 0n, and Жп = 2ln. D Corollary 10 (a) If α and гц are positive η-tuples then (2tnfe«i))a"M<6n(aa;i), with equality if and only if α is constant. (b) [Shannon's Inequality] If ρ and q are positive η-tuples with Pn = Qn = 1 then η η X^bgp* < 5^ft log ft. D (a) Since ж log ж = logxx is strictly convex, see above Example (ii), this result follows from the second part of Theorem 9. (b) Take w = q,a = p/q in (a). A direct proof can also be given by noting that the inequality is just η о < Y^Pi(q%/pi) iog(ft/p»), which is an immediate consequence of (J) and the convexity of ж log ж. D Remark (v) It is easy to see that Shannon's inequality holds with log replaced by log6 for any base b > 1. The inequality is fundamental to the notion of entropy; [MPF pp.635-650], [Rassias pp.127-164], [Mond L· Pecaric 2001]. The concept of comparability can easily be extended to functions; see [HLP pp.5- 6].

Means and Their Inequalities 279 Definition 11 If A is a subset of Rn then the functions /, g : A >-+ R are said to be comparable if either for all а £ A, /(a) < g (a), or for all ae A, /(a) > ρ (a). Example (vii) The quasi-arithmetic Л4-теап and the quasi-arithmetic ЛЛ-теап are comparable if and only if the two functions of 2n variables η η are comparable. Example (viii) The basic result III 3.1.3 Theorem 7(a) just says that for all r e R the function Wl)fi (α 4- έ;ш) is comparable to the function 9JtJ^ (α; υ;) 4- ЯЯ^ (Ь; ш)· On the other hand III 3.1.4 Theorem 9 shows that £Ш^ (ab;u;) is not in general comparable to 9Jt£] (а; ш)9Яп] (& ш) · We сап rephrase III 3.1.2 Corollary 5 as a theorem concerning comparability. Theorem 12 If η > 0, 0 < i < m, then the functions Wl[£o] (Π^ Q_{i); ш) on the non-negative non-zero η-tuples α, υ;, are comparable if and only if 1/tq > Σ™=1 Ι/π, when га ra ot^1 (П«м;ш) < nOTni](aw;«0. (7) i=i г—ι D Taking α& = (1,0,..., 0), 1 < г < т, in (7) implies that l/r0 > Σ™ χ !Λ"*· The converse is immediate from III 3.1.2 Corollary 5 and (r;s). Π Properties of analogous generalizations of the weighted power sums defined in III 2.3 Remark (iii) have been considered; see [Vasic L· Pecaric 1980a]. If α, υ; and M. are as in 1.1 Definition 1 define η SM(a;w) = M-1{J^wiM(ai)), (8) г—ι under the conditions: (i) w_ > l,(ii) Μ :]0, oo[i-»]0, oo[, (iii) either Итж_+оо Λ4(χ) = ooor liniz^o Μ (χ) = oo. Theorem 13 Sm &nd Sat are comparable if either (a) M and λί are strictly monotonic in opposite senses, or (b) Μ and λί are strictly monotonic in the same sense, and λΛ/λί is monotonic. If in case (a) M is increasing, or in case (b) ΛΛ/λί is decreasing then

280 Chapter IV D The proof is along the lines of [HLP Theorem 105, pp.84- 85]. D Remark (vi) Further properties of these sums are given below in 5 Theorems 9 and 10. 3 Results of Rado-Popoviciu Type Inequality 2(6) is, in a certain sense, an ultimate generalization of (GA), and it is natural to ask if certain refinements of (GA), such as were discussed in II 3, have extensions to quasi-arithmetic means. If so they should include as special cases not only the results in II 3 but also those in III 3.2. In this section such extensions are considered, and we also discuss certain particular cases that could have been proved using the methods of the earlier chapters. 3.1 Some General Inequalities Theorem 1 Let [m, M] be a closed interval in R, and Μ : [га, Μ] ι-> R be a continuous, strictly monotonic function, and λί : [га, M] ^R be continuous and convex with respect to Λ4; let α be a sequence, m < α < Μ, and w_ a positive sequence. For all index sets I define а(ЛГ,М;ш;1) = α(Ι) = WjM'(Wtj(α; w) · Then α is sub-additive, that is if I, J are two non-intersecting index sets then a(lUJ) <a(J)+a(J). (1) Further if Λί is strictly convex with respect to Μ inequality (1) is strict unless N(ffli(gr,w) = ^(OJljfew;). IfAf is concave with respect to Μ then (~1) holds. D This is an immediate consequence of (J), and the convexity of TV о M~1.U Corollary 2 Let [m, M] be a closed interval in R, and M,Af,1C,£ : [m, Μ} f-> R be a continuous, with Λ4 and /C strictly monotonic, Λί convex with respect to Λ4, С concave with respect to 1С; let α be a sequence, m < α < Μ, and v_,w_ positive sequences. For all index sets I define β (I) = α(ΛΛ Μ; ν; Ι) - α(£, /С; w; /), where α is as in Theorem 1. Then β is sub-additive, that is If I, J are two non- intersecting index sets p(IUJ)<p(I)+p(J). (2)

Means and Their Inequalities 281 If Μ is strictly convex with respect to Λ4, and С is strictly concave with respect to /C then (2) is strict unless (a) Af(SDt/(a; v)) = λί(OJlj(a\ v)) &nd (b) £(Дг(а;ш)) = £(£j(a;u;)). If λί is strictly convex with respect to Λ4, and if for some a, b Ε Ш, С = αΚ, + b then equality occurs in (2) if and only if (a) holds; and if Μ = αΛ4 + Β and С is strictly concave with respect to /C then equality occurs if and only if (b) holds. D This is a trivial consequence of Theorem 1. D Remark (i) These results, that are almost trivial consequences of (J), include as special cases many of the complicated inequalities discussed in II 3 and III 3.2; [Bullen 1965,1970b,1971b; Mitrinovic Sz Vasic 1968d]. Remark (ii) If in Corollary 2 we take Λί concave with respect to ЛЛ and С convex with respect to /C then (~2) holds and the cases of equality are easily stated. As with (R) and (P) the simplest cases of the above result occurs when I = {1,..., n} and J = {n 4- 1, · · ■, η 4- m}, η, га > 1. Then if as in II 3.2.2 we put α = (αϊ,..., an+m) and α = (αη+ι,..., an+m), and use a natural extension of the notation of II 3.2.2(14) we have the following deduction from Corollary 2. Corollary 3 With the assumptions of Corollary 2, and the above notations, Vn+mN(Wtn+m(o;iw)) - Wn+mC(&n,+m(g:,w)) (3) < (vnAf(mn(a;w)) - WnC(^(a;w))^j + (vmN\Шш{а;, w) - W^/^fe; w))). In particular VnAf(Wln(a;w)) - WnC(&n(a;w)) (4) < (yn-i^(S^-i(a;ty) - Wn-1C{Rn-i{mm)) + [ynM{an) - wnC{an)). Remark (iii) The cases of equality in Corollary 3 are easy to state as conditions (a) and (b) of Corollary 2 reduce to: (a)7 ЛГ(Шп(а;у)) =M{mm(a;v))1 (Ъ)' Cfafaw)) =£(&„(&ш)). The following general inequality extends the Mitrinovic & Vasic result that contains extra parameters, II 3.2.1 Theorem 5; see [Bullen 1970b,1971b]. Theorem 4 Let a, v_, w, Л4, Λί, /С, С be as in Corollary 2, and let μ,ν еЖ, then Шп(лГоМ-1^ХЯ1п(М(а);у) 4- μ) - С о КГ1 (Яп(/С(а); w) + ν)λ (5) <И^n_Yл^oΛ4-1(λ,2ίrг_1(/C(a);7;)4-/i,) -Lor^Vi^jij + ^j

282 Chapter IV where VnwnMoAi-1oC(an) x,= WnVn^ /= Wn /= Wn vnWn M{an) ' VnWn^ ' μ WnS* V Wn^V' If Af is strictly convex with respect to Λ4, and С strictly concave with respect to /C then (5) is strict unless (a) M(an) = \r?~^ %n-1 {Λ4(a); j;) + μ , and (b) /C(an) = 5tn-i(/C(a);u;) + ^'· D Since £ о /С-1 is concave, = Ψη£ о /С"1 (^ (On-i(iC(a);m) + */') + ^Г*^)) > Wn-^o/C-^Sln-^/Cia);!!;) + ^') + wn£(an). Since Af о М-1 is convex WJV ο Μ'1 (mn(M{a)-v) + μ ' < Wn-i-Af о ЛГ1 (λΧ-! (Λ4(α); ν) + μ') + wn£(an). Inequality (5) is now immediate, as are the cases of equality. D Remark (iv) If λ = 1, μ = ν = Q, v_= гц, N = С then (5) reduces to a special case of (4)—the simple case in which the last two terms on the right-hand side are absent. 3.2 Some Applications of the General Inequalities We first state some results of the type considered in II 3.2.2 and III 3.2.4 . Theorem 5 (a) Ifr/t < 1 then μ(Ι) = W/ (ίΗΐψ (α; w))* is a sub-additive function on the index sets. (b) Ifr/t < 1 < s/u then и (I) = Vj (W$\a; v))* - WI(m[?[{a;,<w))u is a subadditive function on the index sets. (c) Ifr < 0 < s then p(I) = (Wl[f\a;v))Vl /(^s](a; w))Wl is a log-sub additive function on the index sets. D (a) In 3.1 Theorem 1 takeλί(χ) = χ\ M(x) = xr, r ^ 0,M(x) = logx, r = 0. (b) In 3.1 Corollary 2 take Μ, λί as in (a), and C(x) = xu, JC(x) = xs.

Means and Their Inequalities 283 (c) In 3.1 Corollary 2 take M{x) = xr, JC(x) = xs and N{x) = C{x) = log ж. D Using 3.1 Corollary 3 we now give generalizations of II 3.2.2 Theorem 8 and III 3.2.4 Theorem 19; the notation is defined in those references and is also used in 3.1 Corollary 3 above. Theorem 6 (a) Ifr/t < 1 < s/u then Wn+m{m[:lm(a;,w))u - yn+m(OTLrlmfelO)f > Wn (<ЩМ fe Ш)Г- Vn (OT^l (a; t/))f+ Wm (ю£] (a; u,))tt- Vm (ю£] (a; tj))4, with equality if and only if: either (i) r/t < 1 < s/u and 3JtJ^(a;i;) = 50Tm (a;?;), and Ж\°](а;ш) = 9Я^(а;ш); or (ii) r/t<l = s/u and W$(a;v) = VR[£(a;v); or (Hi) r/t=l< s/u and m\*](a;w) = OT^ (a;u>); or (iv) r/t=l = s/u. In particular Wn(9j4s1 (a; w))u - Vn[Ж^ (a; t/))' (6) > Wn-^fm^faw))" - V^m^U (a;«))' + K< - «„<£), with equality if and only if: either (i') r/t< \<s/t and 9Ett„l t (a; v) = 97t„L t (a; w) = an; or (1/') r/t < 1 = s/ί and OTt^l^a; г/) = an; or ('iii '^) r/t = 1 < s/t and 9^LSl-i(s;w) = an; or (iv') r/t = 1 = s/i. (b) Ifr < 0 < s thaa (anLrLfes))Vn+m ~ Кг1(«;«)Г" (arCfe*))*" ' with equality if and only if: either (i) r < 0 < s and the conditions in (a)(i) hold; or (ii) r < 0 = s and the condition in (a)(ii) hold; or (Hi) r = 0 < s and the condition in (a) (Hi) hold ; or (iv) r = 0 = s. In particular (Mk'tor" > α.„-,„ ΚίΜΓ-1 (7) with equality if and only if: either (i') r < 0 < s and the conditions in (a)(if) hold; or (ii1) r < 0 = s and the condition in (a)(ii') holds; or (Hi1) r = 0 < s and the condition in (a)(Hi1) holds; or (iv') r = 0 = s. D These follow immediately from 3.1 Corollary 3 using the method in 3.2 Theorem 5. D Remark (i) If v_ = w., τ = 0, s = и = t = 1 then (6) reduces to (R), and if v_= w_, r = 0, s = 1 then (7) reduces to (P).

284 Chapter IV Theorem 7 Let r e Ш, f : M+ \-+ R+ be such that 1/(1 + /) is strictly monotonic, and either l/(l-\-f{x1^r)) is strictly convex when r =£ 0, or l/ (l+f(ex)) is strictly convex when r = 0, and if for all index sets I, we deune (Is = Yl y^ wi X{) 1 + /(!^](α;*)) ^1 + /Ы then: (a) χ is sub-additive; (b) in particular, using the notation of Theorem 6, jr n+m Vn+m V^ Щ ^ 1- l + f^&v)) £? ! + /(*) + т~г п+т * т. ΣΓ .i+/(*e<aii)) ,4ί,1+'<ο·ν' witJb equality if and only ifuJv£(a;,v) = 9Rm (a\w); and, Vn ■^ТТЖУ (8) 1 + /(ЯЯМ (α;υ)) ^ 1 + /Ы ^n - Wn I ^η-ι V^ _ 1 + /(«η) V 1 + (/(2«ίί-ι(α;2)) £ί X + /(βί) ~ и with equality if and only if ап = Шп_1{а;у). D These results follow from 3.1 Corollaries 2 and 3 using M{x) = xr, r =£ 0, Л4(х) = logx, r = 0, Щх) = K{x) = C-\x) = \ D Remark (ii) It should be remarked that in particular cases the exact intervals in which the functions l/(l 4- /{x1^)) and l/(l 4- f{ex)) are strictly convex need not be R+ and this puts some restrictions on the η-tuple a. Remark (iii) Repeated application of (8) leads to η ΛΓ η ST^ wi > Yn >p Wj - Vj έί 1 + /(α0 - 1 + /(α<Γΐ(α;3θ) έί1+ /(*)' with equality if and only if a is constant. If y_ = w, and f{x) = xs, 0 < r < s, (9) reduces to Henrici's inequality, 2 Corollary 3. So (8) is a Rado type extension of Henrici's inequality. Remark (iv) If v_ = w_ various inequalities above are much neater, and the corresponding functions of index sets are not only sub-additive but also decreasing. We now consider some special cases of 3.1 Theorem 4.

Means and Their Inequalities 285 Theorem 8 (a) If -oo < r/t < 1 < s/u < oo, r ^ 0, then wMm^(&w))u - (^)</гаГ'(т4г1 («;!>))') (ю) > WnJ(TtUa;w))U - (^jVKLfei))'), \ vnvvn—1 / with equality if and only if either: (i) r/t < 1 < s/u and an = Tv^l^o/^w) = (^~lWn)1/r^n~A^2l); or (ii) r/t = 1 < s/u and an = Wl^a-w); or (iii) Wn-1vn/ r/t<l = s/u and an = (^~±Wn )1/г9Я|гг11(а; ν); or (iv) r/t = l = s/u. (b) \¥п{^(а;Ш) - {<6n{a;v))VnWn/VnWn) (11) > шп_х{%п-Мш) - (®n-i(&;v))Vn-lWn/VTlWTl-1), with equality if and only if ап = (0п(а;гО) n n п-1<Ш7г (с) IfX>0 then Wn(%n{a-w) - χΥ^<δη(α;ν)) (12) with equality if and only if an = XVn'Vn~1^n-i(oL]2L)· D (a) Take JC(x) = x\ C{x) = xu, M{x) = xr, λί(χ) = xs and μ = ν = 0 in 3.1 Theorem 4. (b) Take JC(x) = C{x) = M{x) = x, M{x) = log ж and μ = ν = 0 in 3.1 Theorem 4. (c) Take K{x) — C{x) = λί(χ) = —ж, M{x) = log ж and μ = ν = log λ in 3.1 Theorem 4. D Remark (v) By taking M{x) = log ж the result in (a) can be extended to cover the case r = 0; and similarly taking Af(x) = log ж the case s = 0 is obtained. Remark (vi) Inequalities (11)—(12), as well as the extra cases mentioned in Remark (v) can all be obtained from II 3.2.1 Theorem 5 by taking special values of λ and μ. 4 Further Inequalities

286 Chapter IV Much of this section could have been placed earlier in either Chapter II or Chapter III but the use of quasi-arithmetic means simplifies many of the arguments. 4.1 Cakalov's Inequality The following special case of 3.1 Corollary 3 is a simple generalization of (R). Theorem 1 If Μ is denned on R^, η > 2 then in order that for all positive η-tuples α and non-negative η-tuples u>, Wn ^ 0, we have the inequality Wn(Kn(a;w)-Mn(a;w)) > Wn^(^n^(a',w) - mn(a;w)), (1) it is necessary and sufficient that AA~X be convex. If AA~X is strictly convex then (1) is strict unless αη = 9Jtn_i(a; u;)· D Taking λί(χ) = /С (ж) = С (χ) = χ, and y_ = w in 3.1(4) gives (1); and the case of equality follows using Remark (iii) of that section. On the other hand with η = 2 in (1) we get, writing hi — λΛ{αι), г = 1,2, WiM 1{bi)+U)2M 1(Ь2) .-ifWibi +W2b2 N W\ 4" W2 ~~ \ Wi -{- W2 < So if (1) holds for all positive а, ш then Μ 1 is convex. D Remark (i) In the case of equal weights this result is due to Popoviciu, [Popoviciu 1959b, 1961a]. Remark (ii) If course if Μ = log then (1) is just (R). Suppose now that not all of αϊ,..., αη_χ are equal, that is a/n is not constant, then (1) can be written &n (a; Ш) ~ Жп (α; w) Wn_x That is to say the left-hand side is bounded below, as a function of a. Further this lower bound is attained if and only if an = 9Jtn_!(a;u;)> and this is not possible if maxa^ < an. In particular the lower bound is not attained for non-constant increasing η-tuples. As a result we could ask if the lower bound could be improved in that case. This problem was first considered by Cakalov, in the equal weight case with Μ = log, when of course Шп{а\ w) = ^n(&); [DI p.4$], [Cakalov 1946]. Theorem 2 If Μ is strictly monotonic, w_ a positive η-tuple and α an increasing, non-constant, positive η-tuple and if η > 2 then W2 W2 " -{Zni&m) -9Яп(а;Ш)) > ш η~λ (Япм^ш) - Ж^а;щ)), (2) Wn - ил ' ' Wn^1 - ил

Means and Their Inequalities 287 provided M. x is convex and 3-convex, or 3-concave, according as M. is increasing or decreasing. D We assume that M~x is convex, 3-convex and increasing; the other case follows similarly. Let D(a) be the difference between the left-hand side and right-hand side of (2) and we have to show that D(a) > 0. For simplicity we will write m = Λ4-1, and Dk(x) = D(a'), where a[ = cti, 1 < г < к and o!i — x, к -f 1 < г < η, ak < χ < αη. Further let Bi(a') = Bi(x) = Bi = %i{M(a/);w)) = М(Т1г(а'-ш)), l<i<n. Then for 1 < к < η - 1, Bn = ^Bk + (1 - ^-М(х))9 Вп^ = тр^Вк + (1 - ^-М{х)У Wn Wn ; Wn-гП Wn-i Since m is 3-convex it is differentiable, I 4.7 Theorem 44(b), and simple calculations give Щх) =M'(x)(Wn^n - Wk) (m' о М(х) - m'{Bn)) — (m ο M(x) - m (Bn-!)) . On rewriting this becomes WnWn-x(Wn - wx)(Wn-x - iuJ V + Wk(Wn - wJfWn-! -Wk)(M(x) - Bk)[M(x),Bn,Bn-1-mf}\ As M~x is convex we have by I 4.1 Theorem 4 (b) that [M(x),Bn; m']i > 0; M-1 is 3-convex so m! is convex, I 4.7 Theorem 44(b), and, by I 4.1 (2*) and I 4.7 Remark(ii), [M(x),Bn,Bn-i\m'}2 > 0; Μ is increasing, by hypothesis, and differentiable since m is differentiable, so M! > 0. Hence D'k(x) > 0, 1 < к < η — 1, ak < χ < an. This implies that D(a) = Dn-1(an)>Dn-1(an-1) = Όη-2{αη-ι) > Dn-2(on-2) > ... > D1(a1) = 0 □ Remark (iii) This proof is an adaption of the proof (Ixxii) of (GA), II 2.4.6; see also proof (xi) of (r;s), III 3.1.1 Theorem 1. Since the means in (2) are almost symmetric, Theorem 2 can be stated as follows:

288 Chapter IV Theorem 3 ИЛ4,ш are as in Theorem 2, and if α is non-constant with maxa^ < an then %iimw) -%Kn(a;w) > λη(2ίη_ι(α;^) -Wln-ii&w)), (3) W2_1(Wn — wj) where λη = ΎΎ™/ΎΎΤ— г, and where j is denned by dj = mina' Remark (iv) It is easily seen that λη > Wn-i/Wn and so, for this restricted class of sequences, (2) is more precise than (1). In general, unlike Wn_i/Wn, the lower bound λη is not attained. Example (i) If λη is replaced by a λ^ > λη there are η-tuples for which (3) would then fail. To see this take Μ = log, αγ = 1 — e, e > 0, α<ι = ... = an = 1, when Μα;^)-0η(α;^) = 1 - (wjWn) - (1 - e)^w" Оп-1(а;ш) ~ 0n-i(a;w) 1 ~ (wjWn-J - (1 - e)"*/w» ' and ase-> 0+ the right-hand side tends to λη. Example (ii) There is no corresponding result for (P). To see this consider the following: αχ = ... = αη_2 = 1, αη_ι = an = χ and let μη > (η — l)/n when ,. an(a)/gn(a) n hm j r-n- = 0. η— (?ϊη_1(α)/βη_1(α))μη This shows that the exponent (n — l)/n that occurs in (P) cannot be improved by assuming that a is increasing; [Bullen 1969d; Diananda 1969]. 4.2 A Theorem of Godunova The following simple theorem has many interesting corollaries; see [Godunova 1965]. Theorem 4 Let Μ :]0, oo|W R be continuous, strictly increasing, with M~x convex and Итх->оЛ4(х) — 0> or —oo; suppose further that α and b are two positive sequences, and that for each n,n > 1, υ/η) is a positive η-tuple with Wkn) = 1, and Σ™=]< w^bn <C,k>l. Then oo oo ^2ЬпШп(а;Ш^)<С^ап. (4) n=i n=i 1 n If С = limn-^oo — V^ bn then the constant in (4) is best possible, η f—' k=i D By 2 Example (iv), OTn(а;ш(п)) < Я„(о;ш(п)) = Σ*=ιω*")α*> η ^ L Hence n=i n=i fc = i k=i n=k oo <C V^ dk, by the above hypothesis. fc = l

Means and Their Inequalities 289 Now let dk = 1, < к < га, = 0, к > га; then from (4) 1 m 1 oo -j m с > - Σδ" + - Σ δ"·Μ_1 (Mi)wln)) ^ - Σ ь·. η=ι η=ι η=ι using the hypotheses on Λ4. This completes the proof. D Corollary 5 The following inequalities hold for all positive sequences α and real numbers p, q > 1. n=i " fc=i n=i CO ,ν·, .. П CO <*) Σ(ν1(Π«·Γ)(-1,/η)<Σ-"; n=i ^n fc=i n=i CO 71 , CO w Σ(^τ^(^)(Σ8ίη^α,)ρ)<27ΓΣ<; n=i k=i n=i oo η со П=1 fe=l П=1 со 1 1 η со „ <" Σ(^(;Σ-)*)<Σ^ 71=1 fc = l П=1 со 1 η со (/) Σ(^τϊ(-!Π^)1/η)<Σ«"· n=i fc=i n=i D These all result from Theorem 4. In (a), (c) and (e) α is replaced by ap, and M(x) = x1^. Then: in (a) take wjj.n) = g Jg ~ ^, bfc = ^Ц-^, ra, fc > 1; in (с) wi = sin— tan—-, bfc = ——-, n,fc > 1; in (e) w) = -, 6fc = n2n «4-1 Kn 1 11 -, n, /c > 1, noticing that — < >^ — —, and use (5). «4-1 к *-^ n(n 4-1) n=k ' In (b), (d) and (f) take M{x) = log ж. Then: in (b) take w^n\b as in (a); in (d) take uAn\b as in (c); in (f) take uAn\b as in (e). (f) Same as (e) but with M{x) = log ж, and the sequence just α. Π (η!)1/71 Remark (i) Since e λ < -———, see I 2.2(a), Corollary 5(f) implies Carleman's inequality, II 3.4 Corollary 16; see [PPT p.231). Remark (ii) Taking M(x) = x1^ or log ж, w^ = , ^ ^^ ? bfc = 1, n,« > 1, „(η) _ Уа ^ (α 4- /?)n where a,/? > 0, a 4- /? > 1, then an inequality of Knopp, a generalization of Carleman's inequality, follows from Theorem 4; [Knopp 1930; Levin 1937].

290 Chapter IV Remark (iii) Another proof of (4) that leads to many other examples has been given; [Vasic L· Pecaric 1982b]. 4.3 A Problem of Oppenheim In this section we consider the following interesting question. If the quasi-arithmetic M-means of α and b are in a certain order when can it be deduced that their quasi-arithmetic Μ-means are in the same order? The question was first studied, and solved, by Oppenheim for the arithmetic and geometric means, in the case η = 3; [Oppenheim 1965,1968]. The extension to general n, and in particular the important condition (6) below is essentially due, to Godunova & Levin, [Godunova L· Levin 1970], who however failed to apply their theorem, Theorem 6 below, to Oppenheim's problem. Another answer to the general case is given in [Vasic 1972]; the discussion below follows that in [Bullen, Vasic L· Stankovic]. Theorem 6 Let α and b be increasing positive η-tuples, η > 3, such that for some m, 1 < m < n, Q>i < b{, 1 < i < η — га; a^ >bi, η - m + 2 < г < η. (6) Let w be another positive η-tuple, and M. : R+ иМав increasing concave function with xM'{x) also increasing. If Kn{Q;,w)<Vln(b;w), (7) and if 0 < а < 1 ™~т then 2tn {M(o);w) - aM (2ln(a; w)) < 2ln (M(b); w) - aM (2ln(6; w)). (8) If а = 0 we do not need to assume that xM!\x) is increasing. If Μ is strictly increasing, M' strictly decreasing, and χλΛ'(χ) strictly increasing, a condition that can be omitted if а = 0, then (8) is strict unless a = b. D If 0 < λ < 1 and Cfc = (1 — λ)α^ 4- АЬ^, 1 < к < η, then c, like α and b, is an increasing positive n-tuple. Further define φ(Χ) = ^n{M(c);w) -aM(%n(c; w)), when (8) is just 0(0) < 0(1). This last inequality will follow if we can prove that ф' > О. Now η η η \νηφ'(λ) = Y^w^h - ai)M'(ck)~a(J2m{bi - α^Μ^—^υ^α). (9)

Means and Their Inequalities 291 If a = 0 the right-hand side of (9) reduces to its first term and since M! > 0 and, from (7), Σ™=1 Wi(bi — di) > 0, we have that φ' > 0. So let us assume that a/0. From the definition of c, η η Y^WiCi> Σ WiCi - Cn-m+i(Wn - Wn-m). i=i i=n—m-\-i Consider first the last term on the right-hand side of (9). Since Μ is concave M! is decreasing, I 4.1 Theorem 4(b), and so, again using Σ™=1 щ(Ы — α;) > 0 : Π 1 П П y-jy *\J>2wi(Pi ~ ai))M' {цГ^™*0*) ^a(^2wi(Pi -^))Л4'(сп_т+1(1 ^Г^)) i=i n i=i i=i n η .. <a(y^Wi(bi -ai))M,(cn-m+1)- ш - Kl^ J 1- ν1Ι^Λ V Wn J η < 2_] щ{Ъ% — аг)Л4/(сп_т_(_1), by the monotonicity of xM'(x)(10) Now we turn to the first term on the right-hand side of (9). η n—m η У^ Wi(bi - a,i)M'(ck) = Σ, wi(bi - a,i)M'(ck) - ^2 wi(ai ~ h)M'{ck) i=i i=i i=n—m-\-i n—m η > ^ wi(bi ~ аг)Л4'(сп_т+1) - ^ m(ai - Ь;)Л4'(сп_т+1) i=i i=n — m-\-i η = 2_,wi(bi — аг)Л4/(сп_т+1), by the monotonicity of M!. (11) From (10) and (11) we see that φ' > 0. D Remark (i) It is easily seen that (~8) holds if Μ is convex and decreasing, with xM'{x) decreasing; and if Μ is convex, increasing and xM'{x) is increasing and if (~7) holds so does (~8). Corollary 7 Let η > 3, 1 < m < n, let a, b be two increasing positive n-tuples that satisfy (6), and let w another positive η-tuple. If s e R and тМ(а]Ш)<мЫ(Ъ]Ш), (12) w then if 0 < а < 1 ^^ and s > 0, /TX^(b-w)\a ^W(a;M)

292 Chapter IV and ifO<t<s, {(т^\а;Ш)У-а(Ш^\аЖ))У/Ь < ((Ш#(Ь;иО)* - а(Ш^(Ь;щ))У/г. (14) If (~ 12 J is assumed and if s < 0 (~13) holds , or if s < t < 0 then (~14) holds. There is equality in (13), (14) if and only if а = b. D This is immediate from Theorem 6, and Remark (i), by taking M{x) variously as xr,\ogx,ex. D Remark (ii) The case η = 3, m = 2, wi = u>2 = гиз, s = 1, α = 2/3 of (13) is due to Oppenheim. By consideration of a special case Oppenheim pointed out that if s = 0 the analogous inequality to (13) with 0n(a; w) replaced by $Ln(a\w) does not hold in general. The correct form is given, as we have seen, by (14). Example (i) In general (8) does not hold if a > 1 — Wn_m/Wn. Let η = 3, wi = u>2 = гиз, 0 < αϊ < α2 < аз,0 < Ь\ < Ьч < b$ Then (6) implies that оц < &i, Ьз < аз and а = 2/3. If s = 1 then (13) says that а1 + а2 + аз < 61 + 62 + 63 τ 4-u 4. л а1а2«з / &ι + 62 + 63 \ Q implies that A = < 1. However if a\ = b\ — 1, a2 = D1D2O3 \ai + a2 + a3 J a — 1, аз = bs = 62 = a, a > 2, then So if a > 2/3 and α is large enough we have that A > 1. This gives the counterexample that is due to Oppenheim. The weakest case of either Theorem 6, or Corollary 7, namely a = 0, gives an answer to the problem of Oppenheim stated at the beginning of this section. Corollary 8 Let Μ : R+ и R be strictly increasing .with M! decreasing. Suppose further that a, b and w. are positive η-tuples, η > 3, with a, b increasing and satisfying (6). Then 21п(а;ш) < %п&Ш) =*► Шп(а;щ) < Жп(Ь;Ш). (15) If M' is strictly increasing there is equality on the right-hand side of (15) if and only if а = b. This corollary is clearly equivalent to the following: Corollary 9 With the notations and conditions of Corollary 8, and ifAf : R+ >—>· Ш is strictly increasing then %i(a\w) <У1п&ш) => тп(ЛГ{а);щ) < Wln (λί {Ь); w). (16)

Means and Their Inequalities 293 Remark (iii) Using Remark (i) we see that Corollary 9 holds if we assume instead that Μ is decreasing, and M! increasing; if instead we assume that M. and M! are increasing then the inequality right-hand side of (15) is reversed. Remark (iv) Bullen, Vasic & Stankovic have given a more general condition than (6) for the validity of (15); but then (16) fails and so this more general result has fewer applications. Corollary 10 Let a,b and w be as in Corollary 8, and s,t el. (a) Ift<s then: ©4s' (έ; ш) => a»4t]teш) < И? Cfc ш); (17) (b) Ift> s then: Ш^(Ь;Ш) < Ш^(а;щ) =► Ш^(Ь;ш) < m^(a;w). (18) There is equality on the right-hand sides of (17), and (18) if and only if а = b. D Immediate from Corollary 8 and Remark (iii), using M{x) = xr,logx or , ex. D Remark (v) The case η = 3, s = 1, t = 0, or s = 0, t = 1, w\ = u>2 = ws of (17) and (18) are those originally discussed by Oppenheim in a very different manner. Remark (vi) Since the larger α the stronger is inequality (8), Corollary 7 is more precise than Corollary 10. Corollary 11 Let α and b be positive η-tuples, not being rearrangements one of the other, but such that their increasing rearrangements satisfy (6), and w_ be another positive η-tuple. Further suppose that mina < minb < maxb < max a, then there is a unique sGl such that (<m[t}{b-w) ift<s, ^l[n] (a; w) < > Ж[$ (b; w) if t > s, [ = m^]lb;w) ift = s. D This is an immediate from Corollary 10 and the continuity of 9Jtj^(c; w) as a function of r. D

294 Chapter IV Remark (vii) A generalization of Theorem 6 can be found in [Vasic &; Pecaric 1982b]. 4.4 Ky Fan's Inequality The following important inequality is due to Levin- son; [Bullen 1973b; Gavrea L· Gurzau 1987; Gavrea L· Ivan; Levinson 1964; Pecaric 1989-1990]. THEOREM 12 [Levinson's Inequality] Let I be an interval in R and Λ4 : i" ь-* R be 3-convex, w_ a positive η-tuple, η > 2, α and b η-tuples with elements in I and satisfying max α < minb, αχ + b\ — · · · = ап + Ьп. (19) Then %n{M(a)]w) - M(9Ln(a;w)) < Vln(M(b);w) - M(Vln(b-w)). (20) If Μ is strictly 3-convex then equality occurs in (20) if and only if a, or b, is constant. Conversely if for a continuous Λ4 : i" i—> Ш (20) holds, holds strictly, for all positive 2-tuples ш, and all non-constant 2-tuples α and b with elements in I satisfying (19), with η = 2, then Μ is 3-convex, strictly 3-convex. D In proving (20) we can assume without loss in generality that α is increasing, when of course b is decreasing. The proof is by induction on n. So first let us consider the case η — 2. Note that if Xi £ i", 1 < i < 5, Xi > Жг+з, 0 < i < 2, the 3-convexity of M. implies that [X0,XI,X2]M)2 > [ЖЗ,Я4,Ж5;Л4]2, (21) and (21) is strict if M. is strictly 3-convex; see I 4.7 Lemma 45 (b); Putting x0 = bi,xi = ^2(b;w),X2 = &2,#3 = а2,Х4 = %>2(а',ш),хь = αϊ, and given (19), (21) reduces to the η = 2 case of (20), with strict inequality if (21) is strict. Now assume the result for all η, 2 < η < m — 1. %ш(М(а)]Ш) -Km(M(b);w) = ^(Vi(^;w)-ViH®;u))) + ^(М(ат)-М(Ьт)) < ]^(м(^т-1(а;Ш))~М(Шт.1(Ь;и,)))+^{М(ат)-М(Ьт)), by the induction hypothesis, < Μ (%n (a; w)) - Μ (Slm(b; w)), by the case η = 2.

Means and Their Inequalities 295 The case where ЛЛ is strictly 3-convex, and the case of equality are easily obtained from this proof. Now assume that η = 2 and that (20) holds when a\ — x, b\ = χ + З/ι, a<i — b<i — χ + 3/i/2, w2 = 2wi = 2. Then (20)reduces to 6h3[x + 3h,x + 2h,x + h,x;M]3 > 0. (22) The hypotheses imply that (22) holds, or holds strictly, for all χ e I and h > 0, small enough; so Λ4 is 3-convex, strictly 3-convex; see I 4.7 Lemma 45(e). D Remark (i) If Μ is 3-concave (~20) holds, Remark (ii) Inequality (20) is just the a = 1 case of inequality (8). Remark (Hi) Pecaric has shown that (19) can be replaced by a\ — b\ — · · ■ = an — bn; [Pecaric 1982]. Corollary 13 Let η > 2, and let α and b be positive η-tuples that satisfy (19), w another positive η-tuple. If s > 0 and t < s or t > 2s; or s = 0 and t > 0; or s < 0 and s > t > 2s then ((ш#]&иО)* - (т[:\аЖ)У)1/г < ((т^\ь]Ш)У - (τ$&ν)γ)1,\ (23) while if s > 0 ®пЫ;ш) < ш[:\а;ш) ,24) Equality occurs in any of these inequalities if and only if a, or b, is constant. D This follows from Theorem 12 and I 4.7 Examples (iv), (v). For (23): in (20) take M{x) = xr, put r = s/t and replace α by as. For (24): in (20) take M{x) = logrr, and replace α by as. D Remark (iv) Corollary 13 is analogous to 4.3 Corollary 7. A particular case of this last corollary is a famous inequality due to Ky Fan; [BB p.5; DI p. 150]. Several independent proofs of this interesting result will be given. The following notation is standard in the discussion of the Ky Fan: if α and w are positive η-tuples, η > 2, with 0 < α < -, then we write Я'п&Ш) =21η(1-αι,.·.,1-αη;:ω), &n(a;w) = 0п(1-<ц,..., 1-αη; w), (25) with obvious extensions to other means.

296 Chapter IV Theorem 14 [Ky Fan's Inequality] If a and w_ are positive η-tuples, η > 2, with 0 < α < - then ®п(тш) < %п(а;ш) ( with equality if and only if α is constant. D (г) This is just a special case of (24); take s = 1 and αι + bi = · · · = ап-\-Ьп = 1. (гг) Noting that f(x) = log ((1 —x)/x) is strictly convex on ]0, ^]; the result follows from (J); [Chong К К 1998). (ш) A very simple proof of the equal weight case can be based on the Cauchy principle of mathematical induction, II 2.2.4 Theorem 7; [BB p. 5], [Dubeau 1991b]. Writing а = 21η(α), when 0 < а < |, then (26), in the equal weight case, is just /ττ αί \1/n а /ч When η = 2, and putting ρ = α\α<ι, (26) reduces to p(l — 2a) < a2(l — 2a). This however is immediate by the η = 2 case of (GA), and the fact that 0 < α < \. Now suppose that (27) holds for a particular value of η > 2 we first show that it holds for 2n. Let αϊ = 21η(αι,... ,αη),α2 = 21η(αη+1,... ,α2η),α = ^п(аь ... ,α2η) then 2η νν -Йг α,· \V" (ni^r-jmr^nni^) г=1 V г=1 г=п+1 <л / ( 2^ ) ( =- ), by the induction hypothesis, V Μ — αϊ/VI — аг/ <- =, by the case n=2. 1 — а Now we show that if (27) holds for a value of η > 2 then it holds for η — 1. Let α = 21η_1(α1,... αη-χ), then: _ n-l 21η(α,αΐ5... ,αη_χ) ( = TT -—-— ) <- ™ ?_1? n ,by the induction hypothesis, VI - α x± 1 - a*) 1 -2L(a, a,,... ,an_i) 1-a This completes the induction, by II 2.2.4 Theorem 7. (ш) [Sandor 1990b] Inequality (26) is an easy deduction from Henrici's inequality, 2(4). In 2(4) putting a^ = (1 — bi)/bi, 1 <i <n, gives (25) with a replaced by b. (iv) Another proof, due to Segre, can be found in VI 4.5 Example(v); [Segre]. D Among the many generalizations of (26) the following are worth noting.

Means and Their Inequalities 297 Theorem 15 If a and w_ are positive η-tuples, η > 2, with 0 < α < - then Ъп(а',Ш) ^ <$nte^). Ъ'п&ш) ~ unfaw)' ®n (α; Ш) - ®п (й; w) < 21η (α; ги) - * --!_<*__!_<. -<(а;ш); 1 1 *Ш «nte) - в;ы βη(ο) - a;(a) αη(α)' with equality if and only if α is constant. Remark (v) The first inequality is due to Wang & Wang, [Wang· Ρ F L· Wang W L; Wang W L &; Wang Ρ F 1984), and as a result the combined inequality, Ъп(д;ш) < ®п(а;ш) < %n(q;w) is called the Ey Fan-Wang-Wang inequality. Remark (vi) The second inequality, an additive analogue of the Ky Fan inequality, is due to Alzer; [Alzer 1988a, 1990e, 1997b; McGregor 1996; Mercer P]. Remark (vii) The third inequality is due to Alzer, [Alzer 1990g]\ it generalizes the equal weight case of an earlier result of Sandor, [Sandor 1991a}: 1 1 < 1 1 #nte™) ?)n(Q]w) %i'n{a;w) 21п(а;ш)' Alzer, [Alzer 1995a], has also proved 1 1 < 1 1 S)fn(a;w) S)n(a;w) ®'n{a;,w) 0η(α;™)' Reference should be made to VI 2.1.2 Theorem 13. Extensions to all power means have also been obtained; [Wang· Ζ L· Chen; Wang-, Li & Chen 1988]. Theorem 16 If a is an η-tuple, η > 2, with 0 < α < - and r, s еШ, г < s, then artM(q) < a^i(a) anilr'(a) - тГ(а) or 9s if and only if \r + si < 3 , and — > — ifr > 0, and r2r < s2s if s < 0. r s In addition we have various extensions of Rado-Popoviciu type due to Wang С L; [Alzer 1990i; Kwon 1996; Wang С L 1980c, 1984b].

298 Chapter IV Theorem 17 If a and w are positive η-tuples, η > 2, with 0 < α < - then < Wn(i^(a;w)&n(a;w) - <(а;щ)0п(а;ш)); ^ W-n-l / νβη-ι(α;^)/<_ι(α;υΟ/ ~ \en(a;u;)/<(a;«z)y Much research has been carried out on this inequality and many generalizations have been given by several authors. There is an important survey article by Alzer that contains interesting related results and a comprehensive bibliography, [Alzer 1995a]. Converse inequalities have also been proved and Alzer has given various Ky Fan inequalities for the pseudo arithmetic and geometric means, see II 5.8; [Alzer 1990q]. For further information see all the above references and also the following: [Alzer 1988b,1989f,g,h,i,1990f,h,j,k,m,1991e,1993a,b,1996b,1999a,2001b; Alzer, Rusche- weyh Sz Salinas; Chan, Goldberg Sz Gonek; Chong К К 2000,2001; Dragomir 1992b; Dragomir Sz Ionescu 1990; El-Neweihi Sz Proschan; Farwig L· Zwick 1985; Gao P; Jiang; Ku, Ku L· Zhang 1999; Lawrence L· Segalman; McGregor 1993a,b; Mercer A 1996,1998, 2000; Neuman Sz Pecaric; Pecaric 1980b, 1981c; Pecaric Sz Alzer; Pecaric Sz Rasa 2000; Pecaric Sz Zwick; Popoviciu 1961b; Vasic Sz Janic; Wang С L 1988a; Wang С S; Wang W L 1999; Wang Sz Wu; Wang Z, Chen Sz Li; Yang Sz Wang ; Yang Υ 1988; Zhang Ζ L; Zwick]. 4.5 Means on the Move The results of III 6.1 have been extended to a certain class of quasi-arithmetic means; see [Boas Sz Brenner; Gigante 1995a]. Theorem 18 IfM: R+ ι—► Ш and α and w_ are positive η-tuples then lim (DJln (a 4- te; w) — t) = 2ln (a; w) £—►00 ' provided Μ satisfies one of the two following sets of conditions. (a) lim M{t) = lim Λ^"1^) = oo; lim 4ίττ = lim ^ A'ffl = 0, ν M'(t + s) _. (AO'M .. . . . f lim —, if . = lim , J ,\ . = 1, uniformly on compacts sets, ois: t-oo M[t) t-oo М-г(р) (b) lim M{t) = lim Л4_1(£) = 0; Μ has the other properties listed in (a), and t—юо t—юо ^{Мм-щ8) = 1> (M-1Y(M(t)(i + e)) = (i + v)(iM-1)OM)(t), where e, η —► 0, as t —► oo.

Means and Their Inequalities 299 Remark (i) It is easily checked that if M(x) = xr,r Φ 0, then Μ has these properties. Further extensions can be found in [Aczel, Losconzi &: Pales; Aczel &: Pales; Brenner &: Carlson]. Theorem 19 Let Μ he a homogeneous function on positive η-tuples differentiable at e, and such that if α is constant, α = (α,..., α), say, then Λ4(α) = α. If дМ/дак{е) = Wk, 1 < к <n, then Wn = 1 and lim (Μ (α + te) - t) = 2in (a; w). (28) If Μ has continuous second order derivatives in the neighbourhood of e, Л4(а + *е) = * + 21п(а;ш) + 0(1Д), t -> oo. (29) D By Euler's theorem on homogeneous functions, see I 4.6 Remark (ii), Σ/ϋ=ι ак~^—(а) = ЛЛ(а). Substituting а = e gives Wn = 1. In addition lim (M(a + te) -t) = lim tiMCb^a + e) - l) ί-юо ч ί-юо ч 7 Л4(е + sa) - Л4(е) ^д,/ч Έ.Έ . , ч = hm —- — — = V.M(e).a, which is (28). Finally if Μ if has continuous second order derivatives in the neighbourhood of e, M{a-\- te) =tM(e + a/t), by homogeneity, дм дак Л А А Ш(е) + У^ак-—(e) + 0(t_1), by Taylor's theorem, fc=l which is (29). D Remark (ii) An interesting application of this result to define scales of means is in [Persson &; Sjostrand], see III 6.1. 5 Generalizations of the Holder and Minkowski Inequalities In section 2 we discussed the comparability of means and obtained the inequality 2(6), a generalization of (r;s). It is natural to ask if the other inequalities between the power means, III 3.1.2(7) and III 3.1.3(8), that are derived from (H) and (M), have analogues for the more general means being considered here. First we give a simple generalization of III 3.1.3 Theorem 7(b) and inequality III 3.1.3(9); [Aumann 1934; Jessen 1931a].

300 Chapter IV Theorem 1 Let Λ4 and Μ be continuous strictly monotonic functions denned on the same interval, then for all appropriate α^ = (a^,... ,ain), 1 < г < m, ati) = (μ1ό,..., amj), 1 < j < n, and non-negative η-tuples и and v_, ^п{Шт{а^-и)-у) < mm^ln{a{i)-v)-u) (1) if and only if the quasi-arithmetic Η-mean, where Η = λί ο Λ4_1, is a convex function on the set of sequences on which it is denned. D (1) is an easy consequence of the property of the function Η stated in the theorem. D Remark (i) A simple condition that implies that the function Η above satisfies the required condition is given in Theorem 7 below. We now consider a more general problem and to fix ideas suppose that we have three functions /C : [/ci,^] »—> ]R, С : [Д,-^] »—► R, Л4 : [7711,7712] ι—► Ш. and the associated quasi-arithmetic means: Яп(а;г/;), £п(к,ш), and 9ЯП (с; w) where υ; is a positive η-tuple with Wn = 1. We are interested in obtaining inequalities of the type f(Ma;w)^n(b;w)) > mn(f(a,b);w) (2) or its reverse, where / : [hi, fe] x [-^1,-^2] l_► [^1,^2] is a continuous function. The most important examples of / are (i) f(x,y) = x 4- y, when inequality (2) is said to be additive, and (ii) f(x,y) = xy, when inequality (2) is said to be multiplicative. For instance the III 3.1.2(7) is multiplicative, while III 3.1.3(8) is additive. The extreme generality of (2) means that often different looking inequalities are in some sense equivalent and it is worthwhile clearing up this point before proceeding to the main theorem. (A) A new inequality can be obtained from (2) by changing / as follows. Let σ : [т^тг] ι—► [ji,J*2] be strictly monotonic and continuous and put д = σο/, J — Mo σ_1, when (2) becomes g(£n(a;w),£n(b]w)) >3n{g(a,b);w), (3) if we assume σ to be increasing, or (~3) if we assume σ to be decreasing. Example (i) In particular taking σ = Λ4, we have that J = 1, and (3) becomes 0(Яп(а;ш),£п(&ш)) > Яп(0(а,£);ш), where д = Μ ο /. (B) Instead of changing / we could change α and b as follows.

Means and Their Inequalities 301 Let к : [ki, k2] »—► [si, s2], λ : |Д, £2) 1—► [t\, t2] be strictly monotonic and continuous, put a* = «(a), 6* = λ(6), and let ρ : [si, S2] x [£1, £2] l—* [mb m2] be denned byg(s,t) = /(«-1(s),A-1(i)) and imaIIyput<S =/Co к"1, Г = £ολ_1, then (2j becomes р(©п(а*;ш),з:п(Ь*;ш)) > ягЦ$(^ь*;ш), (4) or f~4J. Example (ii) In particular taking к = /С, Χ = С then <S = Τ = 1 and (4) becomes p(2in(a*;^),2in(6*;^)) > ^η(ρ(α*, 6*);^), where g(s,t) = f(K-1(s)iС~Щ. Definition 2 Inequalities derived from an inequality of type (2), or (~2), by a finite number of applications of the procedures (A), and, or (B) are said to be equivalent. Remark (ii) This relation is an equivalence relation3. Remark (iii) It is clear that both of the operations (A) and (B) take strict inequalities into strict inequalities; so if the cases of equality are known for an equality then they can be determined for the equivalent inequality. Example (iii) It is not difficult to see that every multiplicative inequality is equivalent to some additive inequality. Suppose that f(x, y) = xy and that the domains of both /C and С are М+ when (2) is just &n(a]w)£n(b;m) > Wtn(ab;w_) Using (4) with κ = λ = log this becomes 6n(a» + Ъп(Ь*;Ш) > 11η(α* + Ь*;ш) where α* = log α, b* = log b, S = /С о exp, Τ = С о exp, U = Μ о ехр. Example (iv) Consider III 3.1.2(7) with q, r, s positive then, as in Example (iii), putting 7i = eq, 72 = er, 73 = es we get the following equivalent inequality; ЯТЦ,η (α*; w) + 9Я72,n (b*; w) > Ш17з,n(a* + b*; w), (5) q That is if we write (7)~(J) when the two inequalities (I),(J) are equivalent then: (i) (7)~(7), (ii) (/)~(J)=KJ)~(/), (iu) (/)~(J) and (J)~(K)=KJ)~(K); see [CS p.559; EMS p402].

302 Chapter IV provided 1/log71 4-1/log72 < 1/log73, 71,72,73 > 1; these means are defined in 1.1 Example (i). The case (a) of equality for III 3.1.2 (7) shows that in the case of non-constant a*, b* there is equality in (5) if and only if 1/log71 -f 1/log72 = 1/log73, and от =Ъ ■ Example (v) The reader can easily check that the additive inequality III 3.1.3(8) is equivalent to the multiplicative, η η η (exP$^iyt(logai)r)(exp^iui(logbi)r) > (exp^w;(loga^)r), r > 1. i=l i~l i=l Theorem 3 A necessary and sufficient condition for (2), respectively (~2), is that the function H{a,t) = M(f(K-1{8),£-1{t)j), be concave, respectively convex. D Use (A) as in Example (i) and then (B) as in Example (ii) to get the following inequality equivalent to (2), Я(Оп(а*;гу),Оп(Ь*;2^) > Ъп(Н{а\ b*);w) which just says that Η is concave. D Example (vi) Let f(x,y) = xy and M—\ then H(s,t) = KL~1{s)C~1{t). If Я is concave then (2) gives the following generalization of III 3.1.2(7). &n(a& w) < #nfe w)£n(fr;w). (6) In particular if H(s, t) = s1^^^ then Η is concave if q, r > 1 and q_1 -f r-1 < 1; if all these inequalities are reversed then Η is convex. In either case we get III 3.1.2(7) with s = 1. To get the general case take H(s,t) = sp/qtP/r, using ρ here for the s used in 1113.1.2(7). Example (vii) If H(s,t) = (s1^ + t^p)p then Η is concave if ρ > 1, see I 4.6 Theorem 40 (d), (e), and (2) reduces to (M). / log 73 log 73 \ Example (viii) Finally taking H(s,t) = expi logs 4- , logi) then Η V log 7i log 72 / is concave provided 1/log71 4- 1/log72 < 1/log73, see I 4.6 Theorem 40 (d), (e), and then (2) reduces to (5). It follows from the discussion of convex functions of two variables in I 4.6 that it is often convenient to assume the existence of continuous first and second order derivatives of functions being used, and not much generality is lost in so doing.

Means and Their Inequalities 303 Corollary 4 If Η is denned as in Theorem 3 and if H^ < 0, or Н!{2 < О, and Н[\Н22 — Ηi2 > 0 then there is equality in (2) if and only if α and b are constant. D Immediate using I 4.6 Remark (iv) and I 4.6 (21). D Remark (iv) If the inequalities in Corollary 4 are not strict other cases of equality are possible. More precision can be obtained but for details the reader is referred to [Beck 1969b]. Example (ix) Consider the case considered in Example (vi) above, that is #(s, t) = sp/qtP/r then: H'A(s,t) = P-£ - l)s*/*-V/'; Я£(М) = ^ - l)s>'4*'r-2; Я&(М) = 2 P_sp/q-ltp/r-\ Hence qr (Н'АЩ2 - H['i)(s,t) = ^(1 -р/д-р/г)*гЫ*-1Н>'г-1. Clearly iip/q > 0,p/r > 0 and p/q + p/r < 1 then H is concave, by I 4.6 Remark (iv). If these inequalities are strict then by Corollary 4 there is equality if and only if α and b are constant. Suppose however that p/q+p/r = 1, as in (H) when ρ = 15 then we have to consider the quadratic form h^H^ -f 2hkH[f2 4- &2#22> and 2 2 in this case this is just 8р/<Цр/г— (h/s _ k/t)2, which is zero if h/s = k/t\ so qr qr then there is equality in (2) if aq ~ br. Corollary 5 If f(x,y) = χ + у, when H(s,t) = M{KL-1{s) + £_1(0)> and if Ε = Κ!/Κ!',¥ = C'/C",G = M'/M" and if all of these first and second derivatives are positive then (2) holds if and only if G(x + y)>E(x) + F(y). (7) D In this case the conditions in I 4.6 Remark (iv) give <^; ^-^<^; ^—T,^+ * )< 1 1 1 1 1 / G(x + y) ~ ВД ; G(x + y) ~ F(y) ; G(x + y) V ВД ' F(v) > " E{x)F{y)' which easily leads to the result. D Remark (v) The functions /C, £, Μ determine the functions E,F,G respectively, and the converse is true. For instance K(x) = с exp(/ —T-rdtjdu, c> 0, χ > a. Example (x) If Ε = F = G then (7) just says that Ε is super-additive, which is implied if E{x)/x is increasing. For instance if E(x) = ex, с > 0 then /С(ж) = ж1+1/с for which the additive form of (2) becomes III 3.1.3(8), taking с = l/(r-1).

304 Chapter IV Example (xi) If in the previous example we take Ε = tan then /C = — cos and (2) leads to the inequality £п(а',ш) 4- £n(b;w) > £n(& 4- k; w_), if 0 < a,b< π/4; the means being defined in 1.1 Example(v). Example (xii) Another simple possibility is to take functions E, F, G constant; Ε = б, F = φ, G = 7 say, with 7 > e + φ. Then JC(s) = xs,£(s) = ys,M{s) = zs where χ = e1/6, у = e1^, ζ = e1/7. In this case (2) reduces to (5). Corollary 6 Iff(x,y) =xy, whenH(s,t) = M{KL~1{s)C~1{t)), and if К'(х) + х1С"{хУ v ' C'{x)+xC"{xY v ' M'{x) + xM"{xY and ifK',C',M',A,B,C are all positive then (2) holds if and only if C(xy)>A(x)+B(y). D The proof is similar to that of Corollary 5; see [Beck 1970]. D Remark (vi) As in Remark (v), a knowledge of the functions Л, Б, С determines the functions /С, £, Л4 respectively, for instance ВД = с^ ехр(^ ^dt)±du Example (xiii) If the functions A, 5, С constants, A = α, Β = /3, С = 7, say, with 7 > а 4- /5 then the multiplicative form of (2) becomes III 3.1.2(7), with p = l/a,qr = l//?,r = I/7. A different application of Theorem 3 is to determine conditions under which 9Rn(a;ui) is convex as a function of a. Theorem 7 If Μ has a continuous second order derivative and if M' > 0, M" > 0, then Wln(a;w_) is convex as a function of α if and only ifM'/M" is concave. D By continuity Wln (a; w) is convex if and only if and only if for all a, b, TL \~2~]Ш)- 2 ' (8) see I 4.5.1, I 4.6. By Theorem 3 with К = С = Μ and f(x, у) = (ж + 2/)/2 this holds if and only if H(s, t) = M[\M~1{s) 4- |Л4_1(£)) is concave. An application of I 4.6 (21), shows that this is so if and only if •М,("^~) If Μ'(χ) M'{y) ч Λ4"(Ξ±") ~~ 2\Л4"(аО Λ4"(ΐ/)/'

Means and Their Inequalities 305 which completes the proof. D Remark (vii) The main results above are due to Beck although less successful attempts to obtain such results had been made earlier by Cooper; see [Beck 1970, 1975,1977; Cooper 1927b,1928]. Remark (viii) Inequality (8) is another generalization of (M), and Theorem 7 has been used by various authors to obtain other general inequalities. Example (xiv) If a^,w_^ are defined using the notation in Theorem 1, each щ^ a positive η-tuple with Y^z=1 Wij = 1, 1 < г < га, then the function m S(x) = M(^uiTln(xa^i)\w(i)))i where η is a positive га-tuple and M. satisfies the hypotheses of Theorem 7, is convex and m This generalizes the result of Eliezer & Daykin mentioned in III 2.5.4 Remark (i); see [Godunova L· Cebaevskaya; Godunova L· Levin 1968]. Inequality (6) generalizes (H) and this particular case of (6) is derived from two inverse functions in either of the two following ways. Let а > 0, and put k{x) = ха,£(х) = k~x{x) = ж1/" and then define 1С, С by either (i) JC(x) = xk{x), C{x) = x£(x), or (ii) /С(ж) = J^k, C(x) = JQ £. In looking for more straight forward conditions for the validity of (6) than the concavity of K~1{s)C~1{t) one might wonder if any other pairs of inverse functions k, ί could be used as above. That this is not the case follows from the next theorem. Theorem 8 Let к : R+ >—>· R+ be strictly increasing, with continuous second derivative, and k(Q) = 0; let £ = к~г and define /C and С by r-X rX either (a) JC(x) = xk{x), C{x) = x£(x). or (b) KL{x) = / k,£(x)= / £. Jo Jo If (6) holds for all a,b then к is a power function and the inequality (6) is just (H). D By Theorem 3 JC~1(s)C~1(t) is concave so from I 4.6 Remark (iv) we have, (/C"1)" < 0, (C-1)" < 0 and for all x, у > 0, ((κ-η'ωμ-^'ίν))2 < 1С-\х)£-\у){{1С-1)'\х){с-1)'\у) (9)

306 Chapter IV Assume now that (a) holds. Then : /С(/С_1(ж)) = /С-1 (х) к (/С-1 (ж)), from which ^~1(χ)) = /c=%)'Hence ^"1(χ) = '(*%))■ °r * = £(^c%)) and s° /C-1(x)£-1(x)=x. (10) Differentiating (10) twice gives (/cr1)"/:-1 + 2(jc-1)\c-1Y + /cr1^-1)" = о (ii) Applying (9) with χ = у we get from (11) that ((/c-1)"/:-1 + /c-1(£-1),/)2 = (2(/c-1),(£-1)/)2 < «r1/:-1^-1)"^"1)" or ((/С"1)"/:-1 - /С-1(/:-1),/)2 < 0. Hence (/c-1)"/:-1 = /cr1^-1)" = -(/c-1)'^-1)', (кг1)"/:-1 + (/c-1)'^-1)' = о, or (/С-1)7/! г is constant. This by (10) implies that — — is constant or /C i(x) /C_1 is a power, from which the theorem follows. If instead we assume that (b) holds then (9) becomes _.,, )mfn. < 1, where apk'(a)i\p) χ = /C(a),2/ = £(/?). Since t is the inverse of к this can be written, putting Ί = c(0) *(<») < _^M_ /с(ж) Since a, 7 are arbitrary this implies that ч is constant and so к is a power.D ж А/(ж) Remark (ix) The part (a) of this result is due to Cooper; see [HLP pp.82-83], [Cooper 1928]. Some results have been obtained for the generalized weighted sums defined in 2(8) Theorem 9 If Μ : R+ f-> R+ is convex, strictly monotonic, with M(Q) = 0 and log oM. о ехр convex then Зм(а + Ъ]ш) <SM(a-,w) + SM{k]w)' This result, due to Milovanovic Sz Milovanovic, generalizes an equal weight case of Mulholland; [AI p.57], [Milovanovic & Milovanovic 1979; Mulholland]; see also [Pales 1999].

Means and Their Inequalities 307 Theorem 10 If Μ has continuous second derivative, is strictly monotonic and convex and if M./M! is convex then <5^(α;υ;) is convex as a function of a. This result is due to Vasic L· Pecaric, and generalizes a result of Hardy, Littlewood & Polya; [HLP pp.85-88), [Vasic L· Pecaric 1980a]. 6 Converse Inequalities In this section the converse inequalities of III 4 are extended to quasi-arithmetic means. Several authors have results of this kind, see for instance [Jzumi], but the discussion below gives the very general results of Beck, [Beck 1969b). Theorem 1 Let Μ and J\f be strictly monotonic functions denned on [m,M], a sub-interval of R, TV increasing, λί strictly convex with respect to Λ4. Further suppose that f : [m, Μ] χ [га, Μ] f—>· R is continuous, strictly increasing in the first variable, strictly decreasing in the second, and with f{x,x) — C, m < χ < Μ. Then for all appropriate η-tuples α and all non-negative η-tuples w_ with Wn = 1, 1(У1п{а;ш),Жп{а;ш))>С, (1) there is equality if and only if α is essentially constant. In addition there is at least one λ, 0 < λ < 1, such that /(^nfe^),OTn(a;^))</(0T2(m,M;A,l-A),OT2(m,M;A,l-A)). (2) If X is unique there is equality in (2) if and only if there is an index set I such that Wj = λ, di = M, г e I, cti = m, г φ Ι. D Inequality (1) is an immediate consequence of 2 Theorem 5 and the properties of /. Since m < а < Μ by 1.1 Lemma 2(c) there is a λ, 0 < λ < 1, such that a^ = 012(^71, M; Xi, 1 — Xi), 1 < i < n. So by simple calculations Wn(a;w) =912(m,M;A, 1 - λ) where λ = 2in (λ; w). (3) If a =m2(m,M;Xi,l - λ»), 1 < г < η, then Wln(c;w) =Wl2(m, Μ; λ, 1 - λ). By 2 Theorem 5, с < α, and so by 1.1 Lemma 2(a) 9Яп(с;ш) ^ 9Rn(a]w_)] that is Жп(а;ш) >Ж2(т,М;Х,1-Х). (4) Using (3), (4) and the properties of / we get inequality (2), for some λ, 0 < λ < 1. Now call the right-hand side of (2) φ(Χ), 0 < λ < 1; then this function is continuous and by the first part φ > С, further 0(0) = 0(1) = С; so for some λ, 0 < λ < 1, φ

308 Chapter IV takes its maximum value and this gives (2) for some λ, 0 < λ < 1, and completes the proof except for the cases of equality. For equality in (2) we need equality in (4), which requires that с = α, or that Wl2(m,M-Xi,l-Xi) =У12{т,М;\г,1-\г), 1 < г < п. Since m ^ Μ the hypothesis of strict convexity of λί with respect to Μ then implies that λ^ = 0, or 1, 1 < г <n. This gives the cases of equality. D Remark (i) Note the essential use of non-negative, as opposed to positive weights. Example (i) The basic examples of / are χ — у and χ/у, when С = 0,1 respectively. Example (ii) Taking f(x,y) = x/y, λί(χ) = xs, s ^ 0, = logrr, s = 0, M(x) = xr? r 7^ 0, = log ж, r = 0, r < s, then the second part of Theorem 1 reduces to III 4.1 Theorem 3; while if we take f(x,y)=x — yit reduces to III 4.2 Theorem 10. In HI 4.1 Remark (ii) an extension due to Beckenbach was mentioned. We show that such an extension exists in the present general situation and state the Beckenbach result as a corollary; [Beckenbach 1964]. For this extension we use the following notation: If 0 < s < η let α = (αχ,... ,αη) = (Ьь ..., ba, сь ..., cn_s) = (Ь,с), w = (wb... ,wn) = (ni,...,ns,^i,... ,vn_s) = (u,v). Theorem 2 Let Μ,λί and f be as in Theorem 1, b an s-tuple, m < а < M, Ш — felO a non-negative η-tuple with Wn = 1, and put σ = 1 — Us = Vn-S. Then for all appropriate c: (a) there is a y, m < у < Μ, such that f{Kn(a;w),mn(a;w)) > /{^а+г(Ь,у;и^),Т18+1(Ь,у]и,а)); (5) if у is unique then there is equality in (5) if and only if for all C{ with щ > 0 we have that Ci = y; (b) there is a λ, 0 < λ < σ such that /(^(а;ш),^п(а;ш)) (б) < /(^s+2(Ь, m, Μ; η, λ, 1 - λ), Ж3+2(6, m, Μ; и, λ, 1 - λ)); if λ is unique there is equality in (6) if there is an index set I such that Wi = λ and di = M, i e I,di = m, г φ Ι. Π (a) Put x = <Ttn_s(c;u>s+i/a,. ..,ΐϋη/σ); then 9ΐη(α;υ;) = 9ΐη (Ь, с; и, υ) = ^δ+ι&^;^σ)5 a trivial identity.

Means and Their Inequalities 309 If ζ = Wln-s(c',ws+1/a,... ,wn/a) then trivially, 9Rn(a',w) = Wls+1(b,z]u,a). Prom the hypotheses and 2 Theorem 5 ζ < x, and so by 1 Lemma 2(b) 9ftn(a;w) = 9fts+i(fr, z\u,a) <Wls+1(b,x;u,a). (7) Hence using the above identity , (7) and the hypotheses /(^п(а;ш),9Лп(а;ш)) > f(Ws+1(b,x-,u,a),Wls+1(b,x]u, σ)), and taking as у the values of χ for which the right-hand side of this last inequality is a maximum, (5) follows. The cases of equality are immediate, (b) The proof is similar to the corresponding proof of Theorem 1. D Remark (ii) Theorem 1, in particular (2), shows that in general the right-hand side of (5) is an improvement on the obvious lower bound C. Corollary 3 Let w_be a positive η-tuple, b a positive m-tuple, 1 < m < n; then for all positive (n — m)-tuples с = (cm+i,..., cn), and any r, s, —oo < r < s < oo, ЯЛМ (Ь,с;ш) _ m£\b,a;w) where, except when r = —s = —oo, α is constant with each term equal to a, minb, ifr = —oo < s < oo, ΣΓ-.«'<*?ν/(ί"Γ) r *-' -i , jf-oo < r < s < oo, I max b, if —oo < r < s = oo; if r = —s = — oo then a is arbitrary, subject to minb < а < maxb. There is equality on (9) if and only if Ci = a, m + 1 < г < η, except in the case r = — s = — oo, when there is equality if and only if minb < с < max b. D Except in the cases that involve r = —oo, and, or s = oo, this is an immediate consequence of Theorem 2(a) with f(x, y) = x/y, and ΛΛ,λί appropriate power or logarithmic functions. Consider as typical of the cases when one of r, s is infinite, the case r finite and а \ max{£,c} not zero, s = oo. Look at the function ф(с) = —τ-, . Simple computations Щ^\Ь,с-щ) show that if Cj < max{b,c} then φ^ < 0, while if Cj = max{b,c} then <//· > 0. So (8) holds in this case with equality as stated. If r = — s = —oo consider instead the function ф(с) = max{b, c}/min{b, c}. Prom the definition of α in this case min{b, a} = min{b,max{b,a}} = maxb. Since in any case min{b,c} < minb and max{b,c} > maxb, the result holds in this case with equality as stated. □

310 Chapter IV Corollary 4 Let 0 < τττ,ι < m2, w_ be a positive η-tuple with Wn = 1, b a positive m-tuple, 0 < m < n. Then for any positive η — m-tuple c, τττ,ι < с < rri2, and any r, s, —oo < r < s < oo, we have жМ(Ъ,с;щ) < ^]+2fembm2;^a-a,g) ^Lr] (b, c; w) " 9я£]+2 (Ь, mi, m2; ν, σ - α, α)' where v_ = (w1,..., wm), σ = 1 — Wm, and α = 0 if θ < 0, α = # if 0 < # < σ, and а = σ if θ > σ, where: (i) if r, s are finite and rs ^ 0 then, θ = _1_ (гЦТ^тЩ + ат:) _ Β{ΣΖι^4 + ση{) s — г \ ra£ — m\ ml ~~ rni (ii) ifr, s are unite and r = 0, or s = 0, θ is given by the appropriate limit of (10); (Hi) ifr is unite, and s = oo, θ = 0; (iv) ifr = —oo and s is unite, θ = σ; (ν) if r = —s = oo, θ = σ/2. Equality holds in (9) for unite r, s if and only if there is an index set I with Wj = σ, q = m2,i £ I, Ci = πΐι, i £ i. If r = —oo , and s is finite, [r unite and s = oo], there is equality if and only if the previous condition holds and minb = 777,1, [maxb = 777,2]. If r = —s = 00 there is equality if and only if min{/3, c} = 777,1 and max{/3, c} = 777,2 · D If r and s are finite this is a consequence of Theorem 2; the other cases can be checked by inspection. D 7 Generalizations of the Quasi-arithmetic Means As general as the definition of a quasi-arithmetic is, even more general means have been defined and we consider some of these generalizations in this section. In the end the extreme generality leads to topics in functional inequalities which are beyond the scope of this book; see [Aczel]. 7.1 A Mean of Bajraktarevic Let / be a closed interval in R, a^ e I, 1 < i < η, ω_ — (ωι,.. -, ωη) an η-tuple of weight functions, where ωι : I 1—► R+, 1 < г < η, and Μ : I 1—► R is strictly monotonic; then define νΚη(α·ω) = Μ г' -— ; (1) V 1^%=ιωΛα%) J as with 1(2) the function Λ4 is said to generate, to be a generator of or to be a generating function of this mean, the Bajraktarevic mean. Example (i) If each ωι is constant, wi say, 1 < г < η, then (1) reduces to the quasi-arithmetic A4-mean, 1(2). (10)

Means and Their Inequalities 311 Example (ii) If ωι = Wi ω, W{ e R+, ω : i" \-+ R+, 1 < г < η, then (1) becomes Example (iii) A special case of (2) is the Gini mean, Ф^'9(а; w) of III 5.2.1, that is obtained by taking ω(χ) = xq, and M(x) = xp~q, or log ж, according as ρ > q, or ρ = q. These general means were first defined in [Bajraktarevic 1958,1963,1969; Daroczy 1964] where 2 Theorem 5 is extended to cover these generalizations. The next theorem due to Losonczi extends 5 Theorem 3; see [Bullen 1973a; Loson- czi 1971b]. THEOREM 1 Let fin(a; к), £nfe A) and ΤΧη(α\ ν) be three means as denned in (1), with common interval I and the generating functions 1C,C,M being differentiable and strictly monotonic, in particular Μ strictly increasing; and let, f : I2 i—> / be differentiable. Then /(£η(α;κ),£η(&λ)) >OTn(/(a,b);i/), (3) if and only if for all u, v,s,t e I, and i, 1 < i < n, Μ ο f(u, ν)- Mo f(s, t) \ (vj о f(u,v) M'of(s,t) J\"n°f(s,t)J (4) If the sign in (3) is reversed or replaced by equality the same is true of (4). D (a) We first prove the necessity of (4). Given г, 1 < г < η, put СЦ = и, b{ = υ and dk = s, bk = t, к =fi i ; then (3) becomes (щ о f(u, ν)) (Μ ο f(u, υ) - Mo f (fin, £n))) (5) η < (.Μ ο/(£„,£„)-Λί ο/(s,t)) Σ Vk°f(a,t), where fin,£*n denote respectively fin (α; κ), £n (b; λ) with the above choice of а,Ь . Using the differentiability of Μ and / we have that Moffat-Moffat) (6) = (Mf о f(s,t) + €l) ((/{(M) + е2)(Лп - s) + (/£(M) + e3)(£n - *)),

312 Chapter IV where 61, 62, 63 tend to zero as (Яп, £n) tends to (s, i). If the definitions of λ^ and щ are extended by putting λ^ = λη, щ = vn, г > η, then it is not difficult to show that limn_^oo &n = s, limn_^oo £n = t\ hence Итп_юо ei = limn-^oo £2 = limn_^oo 63 = 0. In fact: Using these values in (5) and (6) leads to (4). (b) Now we consider the sufficiency of (4). Put и = щ, ν = bi, 1 < г < η, s = Яп(а',к), t = £η(α; λ) in (4) and add the η inequalities obtained. This leads to M(mn(f(a,b);v)^ -Λ^ο/(^η(α;/ς),£η(α;λ)) ^W(*n(a;«),£n(a;A)) <0, which gives (3). (c) Obviously the inequality sign in one of (3) or (4) can be reversed, or replaced by equality, if it is so replaced in the other. Π Remark (i) Let E{ = {(u,v,s,t); (4) holds with equality^ then, from the above proof, there is equality in (3) if and only if (сц, Ьг,Яп{д/) к), £n(b; λ)) £ Е{, 1 < г < п. Remark (ii) Theorem 1 can be extended quite easily to the case of an / : Im \—► /. Example (iv) If κ, λ, ν_ are equal and constant, see Example (i) above, (4) reduces to If we now assume that /С, £, Л4, f all have continuous second derivatives this last inequality can be seen to be equivalent to H^h2 + 2Η"2ϊι1ϊ + Я|2/с2 < 0, for all h, k; where Η is the function defined in 5 Theorem 3. This just says that this quadratic form is positive semi-definite, equivalently Η is concave; see I 4.6 Theorem 40(g). Theorem 1 can be used to obtain a condition under which these means are comparable.

Means and Their Inequalities 313 Corollary 2 Let Яп(а\к) and Шп(а;к) be two means as defined in (1), with common interval I and the generating functions Κ,,ΛΛ being differentiable and strictly monotonic, in particular M. strictly increasing, then for all aeln if for all u,s e I, (M{u) - M(s)\ щ(и) )\щ(и) < /JC(u)-JC(s)\kj(u) 1<-< ' vn(s) ~~ \ /C'(s) /«n(s)' ~ ~ V M'{s) In particular: (a) ify_=K then the means 9Jtn(a; ι/), &η(α;ι/) are comparable if К is increasing, respectively decreasing, and Л4 is convex, respectively concave, with respect to /C; (b) if Μ = 1С and v\ — · · · = vn — ν, κ\ — ··- — κη = κ then the means Шп(а; ν), 9Яп(а; к) are comparable if κ/ν is monotonic. D The main statement is an immediate deduction from Theorem 1; just take f(xi y) — x, and (b) is immediate. As for (a) this follows from the fact if 1С is increasing and M. is convex with respect to /C then {M{u) - M{s))/M'{s) < (JC(u) -/C(s))//C'(s); see I 4.1 Lemma 2 and I 4.5.3 Definition 33(c). D Remark (iii) The argument in Corollary 2 can easily be extended to show that for anyF:/^/, 2ttn(F(a); v) < F(^n(a; к)) if and only if F(u)-MoF(s)\vj(u) /K(u)-K(s)\ Kj(u) Corollary 3 Let Яп(а;к) and Шп{дг,к) be two means as defined in (1), with common interval I and differentiable strictly increasing generating functions, then 9Χη(α]ΐ/) = Яп(а;к) for all α e In if and only if 1С = (aM + P)/(jM + δ) and Ki = €νι{ηΜ + δ), 1 < г < η, where the constants α, β, 7, δ e satisfy the conditions: (i) e(72 + δ2){αη — β δ) =fi 0, and (ii) if 7 ^ 0 then —δ /η is not a value of Μ and a/7 is not a value of 1С. D Using Corollary 2 and the cases of equality in Theorem 1, with f(x,y) = x, the above equality holds if and only if (M(u)-M(s)\Vi(u) _ (1C(u)-1C(s)\Ki(u) ,, \ M'{s) )un(s)~\ K'(8) )*η{8)'-%-η' [) Fixing s = sq e I and putting m(u) = (M{u) — M(sq))/M!(s0), and k(u) = (lC(u) — /C(so))//C/(so) it is easily seen that if u, s £ I \ {so} then 'm{u) — m(s)\ m(t) _ /k(u) — k(s)\ k(t) ra'(s) J m(u) \ k'(s) J k(u)'

314 Chapter IV Now fixing s = si 7^ so we get that K{u) = [aM{u) + β)/{ηΜ(η) + δ), и =£ so; where the constants a, /3, 7, £ are determined from the values of Л4, Л41\ /C, KJ at so, si. The formula holds at so as well, by continuity. Further since /C is not constant a2 + β2 > 0 and αδ — fij =£ 0. If 7 =£ 0 then K(n) — a/7 = (a£ — βη)/' (j2M(u) — £7), which implies the last condition in the statement of the theorem. Substituting for /C in (7) gives \щ(и) J jM{u) + δ V 1/п(гг) / 7Λ^(ί) + δ' ~ ~ ' ^ ' this shows that the left-hand side is a constant, both as a function of г and as a function of u. This completes this part of the proof. The converse is easily verified. D Corollary 4 With the notations and assumptions of Theorem 1 we have, for all α,έ, Ξ In with α + be In, that Яп(а;к) + £η(& A) < Wln(a + b;v) if and only if for all u,v,s,t e I, 'M(u + v) -M(s + t)\ fvi(u + v)\ , . 'JC(u)-JC(s)\ //ч(и)\ (C{u)-C{t)\ (\{y)_ /C'(s) J\Kn(s)J V £'(*) /Un(*), D This is immediate on taking f(x, y) = χ + у in Theorem 1. D Remark (iv) It might be noted that inequality (8) is unchanged if f κ, λ, y_ are replaced by й, λ, Ρ, where for some suitable η-tuple w, h = (wiKi,..., wnKn),\ = Οιλι,...,wnAn), £ = Oii/i,...,ΐϋηι/η). A particular case of Corollary 4 gives, using Example (iii), the following property of Gini means. Corollary 5 If either ρ > q and max{l + q — p} < q < 1, or ρ < q and max{g — ρ, l}<q<l + q — ρ then ®Рпд(а + Ъ;ш) < К'я&ш) + ®РпЯ(Ь;ш); (9) if either p > q and q — ρ < min{l + q — p, 0}, or ρ < q and 0 < q < min{g — p, 1} then (~9) holds. D First let /С = С = Μ. and for some suitable η-tuple w_, and function ω, к — \ = у_= mx). see Example (ii). Then (8) becomes G{u + v,s + t)< G{u, s) + G(v, t) (10)

Means and Their Inequalities 315 where G(x,y) = ((M{x) - M(y))/M'(yj) (ш(х)/ш(у)). In the case of Gini means we further have that M(x) = xp~q, or log ж, ω (χ) = xq, and so G(x, y) = yg{x/y) where /4 i—(*p-*9), ifp>9, git) =<p-q L*logi, iip = q. It is easily checked that the convexity of g would imply (10), and that g"{t) > 0 if t > 0 and p, q satisfy either of the first set of conditions. The second set of conditions imply that g is concave and which in turn implies (~10). This completes the proof. D If we assume that I — R and that M"', ι/ exist then (10) can be solved in a manner that is essentially unique. Corollary 6 If Μ is a strictly increasing twice differentiable function on R and if ω is a positive differentiable function on R then Wn(a + b;wuj) < Mn(a;wu)+mn(b;wuj) (11) for all real η-tuples a, b if and only if ω is constant and A4(t) = а + pt, for some α, β £ R, β > 0; that is to say if and only if Шп(д/, wlu) = 2infe w)· D As in Corollary 5, (11) holds if and only if (10) holds for all u, v,s,t e R. If G(u + v,t)-G(v,t) G(u,0) - G(0,0) и > 0 and s = 0, (10) can be written as —^ ^ у-^- < v ' ; K—L-L. и и Letting и tend to zero we get that G[(v,t) < G[(0,0). A similar argument with и < 0 leads to the opposite inequality and so G[(u, v) = G[(0,0) = a, say. In the same way we can show that G'2(u, v) = G2(0, 0) = β, say. Hence, G(u, v) = au + βν, with a + β = 0, since G(u, u) = 0. That is to say, fM{x)-M{y)^ V M'(y) Juj{y) v y) y ) Putting у = 0 in (12) gives Μ(χ) = Μ(0)+αω(0)Μ'(0)^-. (13) LU\X) On substituting this in (12) we get that хш{у) - уш{х) u{y) - уш'(у) ■■ a(x-y).

316 Chapter IV Again putting у = 0, we get that a = 1; than putting у = 1, ω (χ) = Αχ + В. Since и > 0 we have then that A = О, В > 0. Hence from (13) M(x) = M(Q) + Л4'(0)ж = a + /to, /3 > 0, and the proof is completed by using 1.2 Theorem 5. D Remark (v) Although the main result in this section is due to Losonczi many results involving particular means of this type had been obtained earlier by other authors; see [Aczel L· Daroczy; Bajraktarevic 1951; Danskin; Daroczy 1971; Daroczy L· Losonczi; Losonczi 1970,1971b; Pales 1987] as well as references in III 4.1, 4.2. 7.2 Further Results 7.2.1 Deviation Means Even more general means have been studied by various authors; see [Daroczy 1972; Daroczy Sz Pales 1982; Losonczi 1973,1981; Pales 1982,1984,1988a,b; Smoliak]. We give some results concerning the deviation means introduced by Losonczi; the symmetric means of Daroczy can be obtained as special cases. Let I be a real interval and denote by T>(I) the set of functions D : Ι2 ι—► R, satisfying: (i) for all χ Ε /, D{x,·) : i" f—>· R is continuous and strictly increasing, (ii) D(x,x) =0,xG/; such functions are called deviation functions. Let D_ = (Ζ>ι,..., Dn) where D{ e T>(I), 1 < i < n, and let αι £ i", 1 < г < η, then К : у \-^ ΣΓ=ι Di(di,y) is strictly increasing and K(mina) < 0 < If (max a). So the equation η Y^Di(ai,y) =0, has a unique solution у such that mina < у < max a, called the a deviation mean, or more precisely a D_-mean of a, written Dn(a; D). Example (i) If Di = D, 1 < г < η, we obtain the symmetric mean of Daroczy-, [Daroczy 1972]. Example (ii) If Di(x, у) = Шг{х)[М.{у) — M{x)), 1 < i < n, where Μ is continuous and strictly increasing, we get the mean of Bajraktarevic, 9Κη(α;ω), defined in 7.1 (1). Let D_ = (D\,...) be a sequence of functions in V(I), and J2m = (D\,..., Dm), m > 1; then D_ is called a reproducing sequence when for all α Ε Ik, к > 1, and t e J, we have that: lim™^ m(Dfc+m(a^+w);Dfc+m) - t) = D(t) Y%=1 А(аг, t),

Means and Their Inequalities 317 m elements where g,(k+rn) = (αϊ,... ak, t, t,..., t) and D is a non-positive function that depends on D_. The set of all reproducing sequences will be written ΊΖ(Ι), and we will write D*(x,y) = D(y)Di(x,y). The following result, due to Losonczi, generalizes results in 7.1. Theorem 7 Let I, h and I2 be intervals in R, f : Д χ Ι2 1—► i" a differentiable function, С e 11(1), D_ e 11(h), Ε e ΊΙ(Ι2); then for all α e I?,b £ Ιξ,η> 1, ®n(f(a,b);Cn) < f(Vn(a;Dn),Vn(b;En)), (14) if and only if for all u,s e h, and v,t e I2- D*k(f(u,v)J(s,t))<Dt(u,s)f[(s,t) + D*k(v,t)ti(s,t), к > 1. (15) If condition (15) is satisfied and if Δ^, к > 1, is the set of(u,v,s,t) for which there is equality in (15) then there is equality in (14) if and only if for all k, 1 < к < η, (ak,bk,®n(a',Dn),T>n(b',En)) e Ak. Remark (i) The properties of "means on the move", see 4.5, have been investigated for deviation means; [Pales 1991]. Deviation means have been even further generalized to quasi-deviation means based on the concept of a quasi-deviation function, that is a function Ε : I2 \—> R satisfying: (i) for all (x, t) e I2, ugnE(x, t) = sign(x - t); (ii) for all χ e I, E(x) ·) : i" f—>· R is continuous; (in) for all x, у £ I, x < у, 11—► E(x, t)/E(y, t), χ < t < y, is strictly increasing. If now aeln there is a unique solution of the equation η ΣΕ(αί,ί)=0, i=l called the quasi-deviation mean of α generated by E. Such means have been the object of much study; see [Pales 1982,1983b,1988a,b). 7.2.2 Essential Inequalities In an interesting paper Pales, [Pales 1990b], introduced the following concepts. Given к, к > 2, continuous, strictly increasing functions M^ : i" = [m,M] \-+ R, 1 < г < к, with associated equal weighted quasi-arithmetic means σ^ΰΆ^,.,-Μ^) = {«; ^(^(а),...,!^»)),^/», η G Ν*};

318 Chapter IV This set is called the space of means associated with the functions Л4^г\ 1 < i < k. Its importance is that if for some continuous F : Ik \—► R an inequality that involves all these means, say F(^l}(a),... ,т^](а)) > 0, a G In, η β Ν*, holds if and only if J4«)>o,ueaJ(a<1))...,im<*)); that is the only such inequalities are those that give a description of the space σι{ΰκ£ , - · · >^n )· Inequalities of this type are called essential inequalities for these means. The basic theorem proved by Pales is the following; as the proof uses the Hahn- Banach-Minkowski separation theorem it will be omitted. Theorem 8 With the above notation and putting Μ. = αο + Σί=ι otiM.^ then и e σ/(971^1),..., 9Л^}) if and only if for any choice of α* Ε R, 0 < г < к, к г=1 In the case of the power means the following can be obtained by using Theorem 8. Theorem 9 (a)Ift<s<r and t < 0 < r then aRl(m^\m^\m^) = {(u,v,w); o<u<v< w)}. (b) IfO <t<s <r then a4(ml\m[°\m[£) = {(ΐΛ,ν,ιυ); 0<u<v, Vs{r-t)/r{s-t)ut{s-r)/r{s-t) < w^y (c) Ifr<s and I = [m, M] then ^(OTlT1,^1) = {(и, ν); m<u<v<M,andv < Wl[2s](m, Μ; λΓ(χ))}, where Xr(x) = t Mr-xr nr ψ 0, log Μ — log χ log Μ — log m' ifr = 0. (a) The inclusion a4{mfM:]M:]) Q {(«,«, ω); 0 < u < υ < w)},

Means and Their Inequalities 319 is immediate from (r;s). Now assume that rst =fi 0. Ii0<u<v<w, then by Theorem 8 we must show that ax* + βχ8 + ixr + δ > 0, χ > 0 =>► cm* + /?vs + 7^r + £ > 0. (16) Divide the left-hand inequality in (16) by xl and then let χ —► 0 we see that α > 0, dividing by xT and letting x-^oowe get that 7 > 0. So au* + /3vs + 7^r 4- δ > au* + /3ns + 71/ + £ > 0, by the left-hand inequality in (16). The proofs of the other cases are similar. (b) The proof is similar to that of (a) but uses Liapunov's inequality, III 2.1 (8), in addition to (r;s). (c) (z): aj(Tl^\m^) С {(u, ν); m<u<v<M, andv < Wl[2s](m, Μ; Ar(z))}. If (u, ν) Ε ^/(ОТ^ЯЯ^) then m <u<v < Μ is immediate from (r;s). Further if 0 < A < 1 then Ж[2](ra, Μ; λ) < 9Я[>*](m, Μ; λ), which on substituting λ = Xr(x) gives χ <Wl[2s](m,M;\r(x)), m < χ < M. (17) Now assume that rs ^ 0 and that и = Ш[^] (α), ν = ШТ^](α) , when from (17) / Mr — ar- ar- — rar \ signs(ai)s < signs!-— -ms + ττ Ms I, 1 < г < η. & ν г; _ ь \Mr_mr Mr-mr /' ~ ~ Adding these gives υ < Wl2 (rn, M;Xr(x)). The other cases are similar. (гг): σΣ(Ж^,Ш[8]) Э {(и, υ); m<u<v < Μ, andv <^](т,МАг(ж))}. Again assume that rs ^ 0, and that (u,v) is in the set on the right. By Theorem 8 we have to prove that axr + βχ8 + 7 > 0, m < χ < Μ =^ aur + βν3 + ^ > 0. (18) Take a convex combination of the inequalities obtained from the left-hand inequality in (18) with χ = m,x = M, using as weights АГ(ж), 1 — Аг(ж), to get that aur + /?(9K2S] (™,M; <V(z)))e + 7 > 0. In addition, from (18), aur + /3us + 7 > 0. Take a convex combination of these last two inequalities choosing the weights, 0,1 — 0, so that the resulting coefficient of β is vs gives the right-hand inequality in (18).

320 Chapter IV The other cases are similar. D Remark (i) Part (c) can be used to obtain the converse inequality III 4.1 Theorem 3, and in particular Schweitzer's inequality, III 4.1 (12); [Pales 1990b]. 7.2.3 Conjugate Means Let D be a symmetric, continuous and internal mean of η-tuples with entries in the open interval i" and let φ be a continuous strictly monotonic function on i" then the conjugate T>-mean, by φ, is defined by: ψ \ η — 1 the definition in the case η = 2 is due to Daroczy, who solved the comparison problem for two such means; [Daroczy 1999]. Later Daroczy & Pales showed which of the conjugate arithmetic means are also quasi-arithmetic means and made the above extension to η-tuples; [Daroczy L· Pales 2001a,b]. Their main results are given in the following theorem. Theorem 10 (a) Let φ and φ be continuous and strictly monotonic, φ increasing, then Ща)<Ща) for all а £ I if and only if φ is convex with respect to φ. (b) If η > 3 a conjugate arithmetic mean is a quasi-arithmetic mean if and only if it is an arithmetic mean. D The proofs of these results, and a discussion of the case η = 2 of (b), can be found in the references. D 7.2.4 Sensitivity of Means Given two equal weighted quasi-arithmetic means 9Я, УХ defined for positive η-tuples, where Ш differentiable with positive partial derivatives, we say that Ш has small sensitivity with respect to УХ if ЩдЯП/дхи .. .,дЖ/дхп) < -. η THEOREM 11 The equal weighted quasi-arithmetic mean Ш has small sensitivity with respect to the power mean QJtJp, r G R*, if and only if Μ'" , ,{M"\2 F«2->(^)· Μ v J\M' equivalently if and only if \M' о A4~1\r is convex ifr > 0, concave ifr < 0. D The proof can be found in [Dux &; Goda]. D - end of Chapter IV -

V SYMMETRIC POLYNOMIAL MEANS The elementary and complete symmetric polynomials have a history that goes back to Newton at the beginning of the modern mathematical era. They are used to define means that generalize the geometric and arithmetic means in a completely different way to the generalizations of Chapters III and IV. These new means give extensions of the geometric mean-arithmetic mean inequality In this chapter we study the properties of these means. In addition generalizations of these means due to Whiteley and Muirhead are discussed 1 Elementary Symmetric Polynomials and Their Means lrrhe elementary symmetric polynomials2 arose naturally in the study of algebraic equations, see for instance, [Milovanovic, Mitrinovic & Rassias pp. 7-8,52-58; Us- pensky,1948, Chap.IX]. Consequently many of the results have been known for a long time — the basic inequality, 2(1) below, being due to Newton and Campbell; see [Newton], [Campbell] The properties of these polynomials and the associated elementary symmetric polynomial means3 have been studied in most of the basic references, [AI pp.95- 107; BB pp.33-35; CEpp.516-517; DIpp.76-71',243-244; EM9 pp.95-96,98; HLP pp.44-64], and of course in various papers; in particular see [Bellman 1941; Brunn; Fujisawa; Mardessich; Muirhead 1902/03] where many of the fundamental results are often rediscovered. Definition 1 Let α be an η-tuple, r and η are integers, 1 < r < n, then the rth elementary symmetric polynomial of α is denned by ^]ω = ^Σ!(ΓΚ); α) r ^3=1 ^ We revert now to Convention 3 of II 1; this convention was abandoned in the previous chapter 2 Also called eiemenfcary symmetric functions. We will prefer the word polynomial as the usage symmetric function has a more general meaning in this book. 3 Again, these means are often called symmetric means but that usage has a more general meaning in this book. 321

322 Chapter V and the rth elementary symmetric polynomial mean of α is deuned by [г], Л1/Г ^1Ш=1у^) · (2) Example (i) It is easy to check that: eL (a) = αχ Η \-αη, ejT (a) = αια2 · ■ ■ an, and βη (α) = αια2 + агаз + · · · + αη-χαη. Example (ii) We have from Example (i): ©ίί1 (a) ^(a), 6kw](fl) = gn(fl), (3) 6»J(fl)=</ щ^Т) · Example (iii) If the polynomial cnxn + cn_1xn~1 + ■ · · + cxx + Co, cn =£ 0, has zeros —αϊ,... , — an then cnxn + cn-1xn~1 -\ \-c1x + c0 = cn(x + a1)(x + a2) ■ ■ · (ж + ап) = cn {xn + eLl] (ajx^1 + ej?1 (о)хл"а + · · · + e™ (a)). Hence <£г](й) = —. l<r<n; see I 1.1 Example (i). Example (iv) A simple formula for the tangent of the sum of η numbers can be given using the elementary symmetric polynomials; [Pietra]. tan I y^afe 1 l + Et^i-^^f^tana) ' A different definition of the rth elementary symmetric polynomial mean is often given. In the references [AI p.95; HLP p.51 ) for instance, the definition is яИ(а) = (6И(а))Г = ^. (4) Example (v) *» ,(ffl) = W or ^1^ = ^(й)- (5)

Means and Their Inequalities 323 The justification for (2) is that &£} has more of the properties of a mean than does Bn ; for instance (3) above and Lemma 2(a) below. However some of the algebraic properties are easier to state in terms of Sn , see for instance (10) below. In addition s£ (a), like e£ (a), but unlike Θ^(α), is defined for all real a. For these reasons, and also because sln is traditional we will continue to use it as well авбМ. When convenient the range of r in the above definitions can be extended as follows: eN(a) = 6L0l(a)=s[°](a) = l; eM (a) = ©И (a) = sM (a) = 0, r > n, r < 0. Other notations such as eJT (a), ejjl^a) etc. will be used without further explanation, see II 1.1 Convention 2, and II 3.2.2. The ratios ^ЧЫ = -5#7,1<*<П, *n J(a) have been called Newton means; [Martens L· Nowicki; Nowicki 2001]. We see from (6), Examples (ii), (v) that ^l](a) = 2tn(a), and 9^η1(α) = £η(α). As is seen from Example (ii) the rth elementary symmetric polynomial mean coincides with arithmetic and geometric means at the extreme values of r, r = 1 and r = n. So it would be reasonable to expect that (5^, 1 < r < n, is a scale of comparable means between the geometric and arithmetic means. The justification of this is given below in 2 Theorem 3(b), S(r;s). It is important to note that the elementary symmetric polynomials can be generated as follows, see Example (iii). fc=i k=i fc=o ^ ' П(1 +акх) = ±еЫ(Фк = Σ (ПМ](а)хк- k = i k = i k=o ^ ' (7) k= or, equivalently, (8) Remark (i) It is possible to define weighted elementary symmetric polynomial means; if w_ is another n-tuple, 6W(a5ffl)=(^y/P. (9) is called the rth elementary symmetric polynomial mean of α with weight w. However as the properties of this weighted mean are not very satisfactory we will not consider it in any detail; see [Bullen 1965].

324 Chapter V Lemma 2 (a) The rth elementary symmetric polynomial mean, (5^ (a) has the properties (Sy), (Ho), (Re), (Mo) and it is strictly internal, mina < 6^(a) < max a, with equality if and only if α is constant. (b) If 1 < r < n- 1 then ekr](a) = 41,(0)+anei:ri](a); sM(a) = ^^(a) + Г-апв^\а). (10) The first part of this simple lemma extends results for means defined earlier; see II 1.1 Theorem 2 and III 1 Theorem 2. In addition it helps to justify the use of (5^(a) as the mean rather than Sn (a). The second part states some simple and useful relations that are readily checked. 2 The Fundamental Inequalities The following simple result is so basic that we will give several proofs. Theorem 1 If α is an η-tuple and ifris an integer, 1 <r < η — 1, then (ΒΜ(α))2>5[Γ-ΐ](α)4Γ+ΐ1(«); (1) (eM(a))2 >е^(а)е^+1](а). (2) Inequality (1) is strict unless α is constant. D (г) We first prove (1) by induction on n. If η = 2 then (1) reduces to the η = 2 case of (GA), see 1 (3) and (6), so suppose that η > 2. and that (1), together with the case of equality has been proved for all r and m where 1 < r < m — 1, 2 <m <n — 1. Also assume that o/n is not constant. Ifl <r<n — 1, then by 1(10): s!r4r+l] - (4rl)2 = ^{A + Ban + Gal), where a =((„ - r)2 - i)5[:z?& - ((«- г-жю2, 5 =(n-r+l)(r + i)s^l + (n-r-l)(r-l)st74rXl] " 2r(n - г^Ч!, By the induction hypothesis γ°η —ι/ ^ ΰη-ι ΰη-ι ' V η — ι / ^ йп-гп-1 ' an-i ΰη-ι ^ an-i Λη— ι · \°) This implies that A < -(sJ^L,)2, 5 < 25^5^1,, С < - (s^1)2; and so (air]\2 <r г (*№ η Jr~^\2 (Λ\

Means and Their Inequalities 325 This proves (1) in this case. Now suppose that g/n is constant, a\ = · · · = αη_ι = a ^ an, when (4) becomes an equality since the inequalities (3) all become equalities in this case. But the right-hand side of (4) can be written -ά(*»2(41ί - *n)2=-~M-i])\- - «η)2 < ο, (*]ω)2 * (г+;!1пЛ+1)^'1]^^+11^· (5) which completes the proof. (гг) (2) is an easy consequence of (1) which can be rewritten using 1(4) as ^ ^ (r + l)(n-r + l) r(n — r) Noting that the numerical factor on the right-hand side of (5) is greater than 1 establishes (2). (ш) A direct proof (2) can also be given. 2 t—s r+s A typical term in the expansion of εζ (a)en (en ) is ( TT a2,) ( TT a^); k=i k=r — s + i and this term has coefficient ( I — ( J which is negative. (iv) Of course taking note of 1 Example (Hi) and 1(7) both (1) and (2) can be deduced easily from I 1.1 Corollary 8; see [Campbell; Green; Nowicki 2001]. This proof shows that these inequalities hold for real η-tuples a. (v) The following identity is proved in [Dougall; Muirhead 1902/03]. 1 ^ (2k\ (r, k) (s[r]\ _s[r-i]s[r+i] = i y^ where (r, k) = Z)l Щ=*;-1 °% ) ( П^г-к αί ) (ar+fc -ar+k+i)2, the summation being over all such products obtainable from 0,1,0,2,... ,an. This identity gives another proof of (1). (ш') Another identity due to Jolliffe, [Jolliffe], also gives an immediate proof of (1). V(r-l)!(n-r-l)!/ Vn) n n ) (n-l)X)((oj-oi)C„_!lir.1)2 2!(n-3) v^// \/ s^ ^2 + (r_!)(n_r_i) L·.((°* "α^α* "ae)c»-*,r-2) 3!(n — 5) v-4/·/· \f \r \^< \2 + (r-l)(r-2)(n-r-2) ^ ((<Zi " aj',(afc " a£)(ap " °?) " + ···, yn-6,i—3;

326 Chapter V where Cn_2,r-i is the sum of the products of (r — 1) factors from the (n — 2) factors differing from (jn —ctj), and the summation is over all possible terms of that type; Сп_4)Г-2, СП_6)Г-3,... are defined similarly. D Remark (i) In [Jecklin 1949c) there is another inequality similar to (5). Remark (ii) Theorem 1, in the form of I 1.1 Corollary 8, was originally stated by Newton without proof; [Newton pp.347-349]. The first proof was given in [Maclau- rin]; later proofs can be found in [Angelescu; Bonnensen; Darboux 1902; Durand; Fujisawa; Hamy; Ness 1964; PereVdik; Schlomilch 1858a; Sylvester]. Dunkel has given a very full treatment of the whole topic, [Dunkel 1908/9, 1909/10]. Reference should also be made to [Kellogg]. The inductive proof was discovered independently by Dixon, Jolliffe and M. H. A. Newman, and is given in [HLP pp.53 54]- Cubic extensions of these inequalities have been given; see [Rosset]. Corollary 2 (a) If 1 < r < s < n, then βΙΓ-ι](α)βΜ(«) < e^(a)etl](o), and в^{а)в^{а) < *H(α)*ίΓι](й). where the second inequality is strict unless α is constant. (b)Ifl<r<n — l, and ejT~(α) > &n (a), respectively Sn (cl) > slT (a), then eW(a) > eLr+l1 (a), respectively *M(a) > s^1 («)■ (c)Ifl<r + s<n then 4Τ+β](a) < еИ(a)<4f](«) and £+s](fi) < s(r](fi)sMfe)_ D (a) and (b) are immediate corollaries of (1) and (2) written as skr] (a) ^skr-l](a) , el;\a) ekr"l](a) > —γί , ana r , -, > skr+ll(a) " slrl (a) ' ekr+ll(a) et\a) ' respectively. Then (c) follows from (a) by repeated application: for instance eM(a)eW(fi) > e^(a)e^(a) >■■■> e^(a)e^r](a) = е[^(а). a

Means and Their Inequalities 327 Remark (iii) Put a[l] = s[n\a)/s[£~l](a), 1 < к < п. Then by Corollary 2(a) di > a^ > · · > Q>n , and if a is not constant these inequalities are strict. Further Π£=ι ak = TYk=iak- Now iterate this procedure using a^.1 , 1 < /c < η to get for each к a sequence aj, , m = 1, 2,... and /с, Итш_юо aj, — ^n(&); see [Stieltjes]. Remark (iv) If eJT (a) < eJT (έ) then one can ask for what functions / is it true that ejT (/(a)) < ejT (/(έ))? This question has been answered and in particular this is the case if f{x) — xp, 0 < ρ < 1; see [Efroymson, Swartz Sz Wendroff]. We now establish the elementary symmetric polynomial mean inequality, S(r;s), the basic result of this section, giving several proofs. Theorem 3 Ifl<r<s<n then (a) <5](а)<<г]Ы; (b) е^(а)<е^(а\ S(r;s) with equality, in either inequality, if and only if α is constant. D (a) This is just a rewriting of (1) using the definition of Newton means given in 1. (b) (z) If 1< m < r then, from (1), (skw~l]skw+l])W < (*[™]) ™; multiplying all of these over m gives (*кг+1]) <(*!i]) or eir+li<eM, (6) which clearly implies S(r;s). The case of equality is immediate, (гг) It is of interest to see that S(r;s) can be proved from the weaker inequality (2). The method of proof is similar to Crawford's proof of (GA), II 2.4.1 proof (m). Suppose that 0 < аг < ··· < αη, аг ^ αη, and define a new η-tuple b by: Ъг = е]г](а),Ък = ак, 2 < η < η - 1 and bn chosen so that 6^(6) = e[^](a), or equivalently en (a) = eJT (έ)· If then we can prove that for any s > r, ©η 00 > ©If (α) the result follows as in the proof of (GA) mentioned above. Let с = (α2, ·.., αη-ι) = (р2,..., bn-i) then by the definition of b, e[[](a) =a1anel[,r22l(c) + (ax + an)ei[r2l](c) + е£!2(с), ekr](a) = ekr]№) =ЬАе{ГГаа](с) + (Ь, + ^)е£га1](с) + е^2(с). Hence, (bibn - aian)ej[r22](c) = -(bi + bn - ai - an)e|[r2l](c), (7)

328 Chapter V or (he[::;](c) + е£:?(с)) bn = aianeJ[r22l(c) + («ι + an - h)e^](c). Since by internality, 1 Lemma 2, an > b1 this last identity shows that bn > 0. Now eW(b) - el?1 (a) =(Mn - а1ап)(е[г5Г22](с) + (Ь, Η- Ьп - a, - On)et'al](c) The second factor is negative by Corollary 2(a). As to the first factor we have from (7) that: sign^ + bn — ax — an) = sign^an - ЪъЪп) = sign(b1(b1 + bn-a1- an) + axan - ЬхЬп) = sign((b! - ai)(b± - an)) = -1. This proves that e[n\b) > e[n\a), which is equivalent to 6^(έ) > β^α), and the proof is complete. (ш) A direct proof of inequality (6) has been given in [РегеГсИк]. For simplicity let us use the notation: ek = ek (а), (ек)г = e\ = eki^a^), e^ = (e\)j, etc. Let б > 0, and consider those a for which er = e, and find for which of these er+1 has a maximum value. We show that this is a unique maximum occurring when a is constant, a say. Then if this is the case Дел and e,+1<[er+1]max=(r;i)a^ = A^Jz_e^/r. By 1 (2) this is (6) and because of the uniqueness of the maximum we also get the case of equality. Let u(a) = er+1 — X(er — e), when ди/дак — e% — Ае£_15 1 < к < п. For a maximum we must have ди/дак = 0, 1 < к < η, so λ= -γ4 1 </c<n. (8) This implies that λ = v^nfe~1fer . The last identity shows that λ is a symmetric Ση function of a, and so in particular is invariant under interchange of any pair ai and dj, 1 < г, j < n. Take /c = 1 in (8), and use 1(10) to get Λ"4·Λ+α^2~5 + «^' У' " " '

Means and Their Inequalities 329 where А,В and С involve neither a\ nor a^. Interchanging a\ and a^, and using the above noted symmetry of λ, we get that A + akB A + a,B 2 ¥Т^С = вТ^С> °r' («i-«*)(AC-B)=0. (9) Hence, reordering α if necessary, there are two possibilities: either a is constant, or for some г, 1 < г < η — 1, αχ =£ α-,, 2 < j < г + 1 and αχ = α,-, г + 2 < j < η. Suppose the second case holds; then from (9) we have that if2</c<z + l then l,fc 1>^с AC - B2 = 0, or %r = Ц=^, 2 < к < г + 1. Hence „l,fc l,fc er_2 °Г—1 ^Г —2 A=-^V, 2<fc<i + l. (10) ^r—ι Now repeat the above argument taking к — 2 in (10), and use 1(10) to get e/J, +afcerL2 and so, since we are considering the second case ——r = r~17 , 3 < к < г + 1. ' ° 1,2,/e 1,2,К ' — — ρ ρ °Г—1 °Г —2 ei,2,...i+i If г < η — г — 1 this process can be repeated until finally λ τι — ι — ν αΐ5 which by the symmetry of λ is a contradiction since a± =fi a2,. · -, сц+г- r Η r = η — m > η — i — 1 this process leads to 1,2,. ..,771 1,2,...,771 + 1 , 1,2,... ,771 + 1 * °r —ι er—ι ι "-m+i^r—2 This is again a contradiction since ax =£ am+1. Hence du/dak = 0, 1 < /c < n, implies that α is constant, α say. It remains to prove that it is a maximum of er+i subject to er = e. Simple calculations show that du = J2™=1(e + (λ — ai)elr_1)dai, and that η η η d2u =22e1r_1d2ai +^Д(^ — аг) }^егг,3_2аа^)ааг n-i r-2 a 1 (fa - 0 Σ d2ai + n(r - 1) Σ da*)2) < 0. r(r — 1) This completes the proof. □

330 Chapter V Remark (v) The inequality S(r;s) is another extension of (GA) and proofs of S(r;s) give further proofs of (GA). S(r;s) was first stated by Maclaurin; it was also proved, probably independently, by Schlomilch and Dunkel. Proof (гг) above is in [HLP p.53]. Muirhead states that Schlomilch gave precedence to Fort who had given a proof in the eighteenth century; see [Brenner 1978; Dunkel 1909/10; Kreis 1948; Maclaurin; Muirhead 1900/01; Schlomilch 1858a]. Remark (vi) It was pointed out by Bauer and Alzer that using 1(3) and 1(5) inequality S(l;n-1) is just Sierpinski's inequality, II 3.8 (58); [Alzer 1989b; Bauer 1986a]. Remark (vii) The second proof of (b) applies with no change to the weighted mean defined in 1(9) provided the η-tuples α and w. are similarly ordered; see [Bullen 1965]. Remark (viii) Inequality in Theorem 3(a) is a generalization of (HA). In the case η = 3 S(r;s) can be given a simple geometric interpretation as follows. COROLLARY 4 Let а\,ач, а3 be the sides of a parallelepiped, and A,B,C the sides of a cube of the same perimeter, area and volume, respectively, then: A> В > С, with equality if and only if the parallelepiped is a cube. D This follows from S(r;s) because A = 6^ (α), Β = б j,21 (a), С = &ψ (а). D Remark (ix) This last result can be extended to η-dimensions; see [Jecklin 1948a]. Remark (x) If ж, у, A are the η-tuples defined in II 5.6 Theorem 16, then it has been proved that 0n(y) < 6^ (A) < 2ln(x); see the references given in that earlier section. As a corollary of S(r;s) we get a proof of an invariance property of the elementary symmetric polynomial means ; see II 5.2 (c) for the same results for arithmetic means. COROLLARY 5 The sequences α and 6^ (a), (52 (й)> · · · increase, strictly, together. D By 1(10) we have that s^+1 = —s%} 4 an+1s[[~^, or equivalent ly 5[Г] л -sl'J = r [r_4/ skrl \

Means and Their Inequalities 331 [r] From S(r;r+1), (6), we easily then see that -^-r- < 6^ < Stn. Hence an UU-sW >^j4-1](an-*n)>0, by the internality of the arithmetic mean. Further if α is strictly increasing the above inequalities are strict. Π The inequality S(r;s) implies that if 1 < r < η then 0n(a) < 6^(a) < 21(a), and it is natural to ask if the outer means can be replaced by other by other power means. Shi has shown that if 1 < r < η then for some ri, Г2, 0 < r\ <r<i < 1, m^(a)<G^(a)<m^(a); further if η = 3 then ri = 0 is best possible, and r<i = 2(1 — log 2/log 3) is best possible; [Shi]. In addition we can ask for an r, r — г^(а), 0 < r < 1, such that ΰΤνζ} (а) = 6^J (а)? Using Theorem 3 we easily see that 1 = r1(a) > · · · > rn(a) = 0. A lower bound is readily obtained since if a = (1,0,..., 0) then rk(a) — 0, к > 2. On the other hand it has been shown that if к > 3 then r (a) < /;lQg(n/(n-l)) _ — log (n/(n — k)) Further this upper bound is sharp, being attained when a = (1,1,..., 1,0); [Kuczma 1992]. Mitrinovic has obtained the following interesting generalizations of inequality (2) and Corollary 2(b); see [Mitrinovic 1967b,c,1968a]. [We write Ake[n~u] for Akc„ where cv = вп ] Theorem 6 Let α he an η-tuple and r an integer, 1 < r < n, then (A^eiT^)2 > (А/уе[гг-/у-1])(А/уе[гг-/у+1]), 0 < ν < r - 1. (11) If in addition {-1)ρΑρβ^-ν+ι] >0, 0<p<v<r, (12) then (-l)vAve^ >0. (13) Π For (11) modify proof (iv) of Theorem 1 by applying I 1.1 Corollary 8 to the polynomial (x — Ι)" ΠΓ=ι(χ + α*)' ηο^ηβ 1(7)·

332 Chapter V Inequality (13) is proved by induction on v, noting that if ν — 1 the result is just Corollary 2(b). So assume the result has been proved \ <v <k - \. From (11) (Δ*-1^"*4"11)2 > (Δ*-4Γ~*])(Δ*~4Γ~*+21), which by the induction process , (13) with ν = к — 1, is equivalent to Afc-l Jr-fc+i] Ak-iJr-k+2] Hypothesis (12), with ρ = ν — к is equivalent to (-IJfc-Mfc-ie^-fc+a] > (-ijfc-Mfc-iej^-fc+i]. (15j and the right-hand side of (15) is positive, by (14) with ρ = к — l,v = k. Hence from (15) the right-hand side of (14) is greater than 1; this of course makes the left- hand side of (14) greater than 1. Using this and (15), and reversing the argument implies (13) with ν = к, and completes the induction. Π Remark (xi) If ν = 0 (11) reduces to (2), and as we noted above if ν — 1 (13) is just Corollary 2(b). Remark (xii) Clearly similar results hold with Sn (a) replacing en (a) Remark (xiii) By considering polynomials of the form Πj=i (x ~ aj) ΠΓ=ι (ж+αΌ then I 1 Corollary 8 can be used to obtain even more general results; [Mitrinovic 1967a]. For instance applying 1 1 Corollary 8 to {χ — α) ΠΓ=ι(χ + αΌ we Set that (е|Г1] - аекг-21)(екг+1] - ае[Г]) < (elT1 - ае!Г"1])2. That is, for every a a2(e|r4rl - (екг_1])2) +«(4Г-ЧГ] " 4Γ"4Γ+ι]) + (£~4+ι] ~ №? < 0. By (2) the coefficient of a2 is negative and so (elT4rl " e!r4r+l1)2 < 4 (ekr~4rl " (eiT"1)2) (£~4+l] ~ (4Ψ) ■ Corollary 7 If (а) (-1)" A? e^I"^ Ю > 0, (b) (-Ι^Δ^ΓΓ^α'ι) > °> and (c) (-Ι^Δ^Γ^α/ι) > 0 then Δ4Γ-,"ι1(α)Δ4Γ-,'+,1(β) " (Δ^^-Ίία))2 < A4rrrl](ai)A4rrr+ll(fii) " (A^etr1^))2. (16)

Means and Their Inequalities 333 D Put ax—x and denote the left-hand side of (16) by f(x), when (16) becomes f(x) < /(0). Simple calculations show that /"(*) = 2(δ4ΓΓΓVi)A4rrVi) - (Δ4ΓΓΓΐ](α1))2) < 0. by (11). Hence f'{x) < /'(0), so that the result will follow if we show that /'(О) < О. Again, simple calculations show that /'(0) = A^rrVjA-eLT+Vj - ДЧГГГ VjA'etTVj· The hypotheses, together with (11) imply that <(Δ4ΓΓ1"](αί))2(-1)"Δ4ΓΓΓ2](«ί) <(-irA4rrr](a;)(A4rrr-1](ai))2, which gives A^rrVjA'e^rVj < А^ЛОАЧ™1^)· Substi- tuting this in /'(0) proves that /'(0) < 0 as was to be shown. D Remark (xiv) Inequality (16) can be strengthened if a[ is replaced by α^,Ι <г < η, and then taking the minimum over all i. Corollary 8 If Ape[n~p] > 0, 0 < ρ < m, then Awekr~w+1] > 0. D The case m = 0 is trivial and the case m = 1 is just Corollary 2(b). Assume that Ape[n~p] > 0, 0 < ρ < ra - 1 implies that д™"^"™4"11 > 0 and we prove that if also Ame[n~m] > 0 then Ame[Tw+1] > 0. Prom (11) (A^eiT™^)2 > A^elT^A^eir™^, which by the induction hypothesis is equivalent to лт-ijr-m+i] Am-ijr-m] ^ (Λ l\ Дт-1е(г-т+2] ~ Am-ig^^1] " ^ ' The inequality Ame[n~m] > 0 is equivalent to Δτη~1βίΓ~τη1 > Aw-1ekr~w+lj, or by the induction hypothesis to ^m-iJr-rn\ — Ρ-ΊΓ > 1- № So, from (17) and (18), *£—— > 1, or А^еГ™^ > А^еГ^21, which is just Amen~m+1] > 0, as had to be proved. This completes the induction. D

334 Chapter V 3 Extensions of S(r;s) of Rado-Popoviciu Type Since the basic inequality S(r;s) is a generalization of (GA) is natural to ask whether generalizations of Rado or Popoviciu type are possible. A fairly complete analogue of (P), II 3.1, due to Bullen, [Bullen 1965], is given below in Theorem 2, but extensions of (R) are much more incomplete. Unlike the similar extension of (r;s), III 3.2.3, our present results cannot be deduced from a general result, as in IV 3.2, but must be proved separately. However the techniques used are the two basic ones used to prove II 3.1 Theorem 1— the use of elementary calculus, and the use of the basic inequality, S(r;s) in this situation, on suitably chosen sequences. As in II 3.2.2 if a — (αϊ,... ,an+m) let us write a = (an+i,... ,an+m), and the put Cjyi = Cm \Ql) ι 6tC. The following simple lemma extends the identities 1(10). Lemma 1 (a) Σ*=οe™ ей, ifs < min{n, m}, En-\-m—s [n—t] ~[s—n-\-t] -r . r Ί t=0 eln em J, if s > max{n,ra}, t=0eln em\ ifm<s<n. (b) If 1 < s < η + m, и — max{s — n,0}, r = min{s,ra}, and if η \ ( s X(s. t) = -*—/ , \——, 0 < t < s. s then t=u D (a) follows immediately from 1(1), (7) or (8) , and using 1(4) it is easily seen to imply (b). D Remark (i) In particular if αη+ι = · ■ · = an_(_m = β then (1) reduces to *lsL = ί>Μ)4τν, (2) t=u and if in addition a\ = · · ■ = an = a, *181п = Еамк^, (з)

Means and Their Inequalities 335 Remark (ii) Identities 1(10) follow as particular cases of this lemma; take m = 1 and replace η by (n — 1). We now prove the main result of this section. THEOREM 2 Let a be an (n + m)-tuple, r and к integers with 1 < r < к <n-{-m, and put и = max{r — n, 0}, ν = min{r, m}, w = max{/c — n, 0}, χ = min{/c, ra}. (a) If ν < w and r — и < к — χ then (b) Ifv>w then ~W\ eisuW -Utw' ' (6) D (a) Rewrite (4) as L < R where d rn+m/ By (1) and S(r;s) t=u > (J2^M^~u]){r~t)/{r~u)(^)t/v)k- (6) t=u This inequality is strict unless a\ — · · · = αη+τη. In certain cases this step is vacuous; in particular ifr = l,/c = n-fra, when all the means in (4) are either arithmetic or geometric means. It follows from (3) that the last expression is the (kr)-th power of the rth elementary symmetric polynomial mean of b where 1 \{sM)1/v ifn + l<i<n + m. Hence by S(r;s) and (3) again (4rL)fc > (ЕА(*;>*)(яГ»1)(*-*)/('-»)(5М)*/'')Г t=W X = isir-u]y(k-x)/(r~u) i^iv]\wr/v /V^ w^ t\fs[r-u]\(x-t)/(r-u)s~[v]\(t-w)/v\r_

336 Chapter V This inequality is strict unless θ£~^ = Ёт. If the previous application of S(r;s) had not given a strict inequality, then neither can the present application. However if, as noted above, the previous use had been vacuous, then strict inequality could occur here. From this last expression we have that R > S, where s = (ELW х(к,щ5£-и]){х-^(г-иН&*-ш)/УУ- In a similar way using (1) and S(r;s), (*i!L)p = (f>(M>iMiiS)P < (£ A(M)(4fc-*¥fc-t)/(^W)i/№)r, t=w t=w (7) the inequality being strict unless a\ = · · · = an+m. So (s[*]+my < ($-χ]η^]γ(Σ Kk,t)^ltx]){x-mr-x}(^]){t~w)/w)r, which gives L < Τ where, Τ = (%%=*, λ(/ο,ί)(^~χ])(χ~*)/(^χ)(^])(^ω)/υ;)Γ· But by S(r;s) and the hypotheses in (a) T < S, and this inequality is strict unless ν = w and r — и = к — χ, or a\ — · · · = an + m. This completes the proof of (a), (b) The proof is similar except that when S(r;s) is applied in (6) and (7) it is applied to the second term only, that is to JsJJJ. □ Remark (iii) Although the cases of equality were not stated in the above theorem, the proof is detailed enough for them to be obtained in any particular case. Remark (iv) If r = l,k = η + m then (4) reduces to the equal weight case of II 3.2.2 Theorem 8(b); while if к = s 4-1, η = 1, m = q, r < s (5) reduces to eH,fe, у" г r *ων (8) a direct generalization of the equal weight case of (P). Results similar to (8) can be obtained for the elementary symmetric polynomials; see [Mitrinovic Sz Vasic 1968e]. Theorem 3 If α is an (n + l)-tuple, and if r, s are integers, 1 < r < s < n, and ifQ<p<q then, екг](й) N'W eif](a) У fq.

Means and Their Inequalities 337 (e[rL )p D Suppose that 0 < ρ < q, and putting χ = an+1 , and f(x) = —?Φ^— then, (4βίι)« by 1 (10), f(x) = l" + " / and f'(x) = Щ^ A where (eM+zeiT11)* (eLl)^ A =((p - ^xejT-11^-1' +P4f4r_1) " rfe^"1) [s] [r—i] _ [r] [s —i] 1 (i-p)ekr-1,elf-1] / РМРИ />-ч >~1J\ \ So, using 2 Corollary 2(b), /' < 0 if χ > 0. hence /(αη+ι) < /(0), which is just (9). The case of ρ = q has a similar but easier proof. D Theorem 4 (a) If α is an (n 4- 1)-tuple, and if s an integer, 1 < s < n, and if 0 <p < q, then, (ЩтУ * ί^^"(4"%)Γ'|^- (ю) V eh1 {а) / <?4 ек J(a) (b) If α is an (η 4- 1)-tuple, and if s, /с, m integers, λ =£ 0, with: 1 < s < /c < n; l<ra<s<n if λ>0; or l</c<s<ra<n if λ<0. (11) then в&Ы-Ав&Ы eW(a)-Aekml(a) e'sl(a) eifl(a) If instead of (11) we have that l<m<s<k<nif\<0;orl<s<k<n, l<s<m<nif\>0, then inequality (~12) holds. (c) If α is an (n 4- l)-tuple, and if s an integer, 1 < s < η 4- 1, and if μ > 0 then s —ι (β^(α)+μ)* > V ~ ek5-1J(q), ^eLl(a) - (s-l)s-ie[r1](o) ' with equality if and only if an+i = ( μ 4- ejj1 — 5—r^- s ~ 1 V e[n ' Remark (v) Ifp = s=l,g = n then (10) reduces to the equal weight case of (P); while if s = η 4-1 in (13) this inequality becomes the equal weight case of the result in II 3.2.1 Example (ii). While Theorem 2 can be regarded as extending (P) to elementary symmetric polynomial means no such complete extension of (R) is known. The following result is a partial extension; [Mitrinovic Sz Vasic 1968b].

338 Chapter V Theorem 5 If a is an (n 4- l)-tuple, and if r an integer, 1 < r < η + 1, and if λ > 0 then (n + l)(2in+1(a)-A6Lrl(a))> (14) n(*n(a) - ХгПг-гЛп + 1){г-1)в1г1] _ n + l-r (6ξ(α)Γ \ witJh equality if and only if an+1 = (n+ l)(2tn_|_i — AS^J. D Putting let /(ж) denote the left-hand side of (14). On simplification, using 1(10), f(x) = n%n+x-(n- 1)λ (n + 1~TsP + -^-s^1^) . \ n + 1 η 4- 1 / ι ι _ И So /(ж) is defined if χ > — ( J f ^ ,, and is easily seen to have a unique minimum at x0 = ILI-Ia1^-1)©^ - П + 1 Г ^7" ■ On substituting, f(xo) is the right-hand side of (14) and this completes the proof. D Remark (vi) With r = η 4-1 (14) reduces to the equal weight case of II 3.2.1(10) with μ = 1. Remark (vii) It would be of interest to generalize (14) by replacing 2ln_|_i by axis] . ©Jl^_1 m some way. 4 The Inequalities of Marcus & Lopes In the case of power means and certain more general quasi-arithmetic means there are inequalities connecting the values of these means at the η-tuples a, b and α 4- b; see III 3.1.3 (8) and IV 5. We now consider such an inequality for elementary symmetric polynomial means. The basic result is inequality (5) below due to Marcus & Lopes; it is the corollary of a more general inequality (1), due to McLeod, and independently Bullen & Marcus; see [Bullen Sz Marcus; Godunova 1967; Marcus L· Lopes; McLeod; Yuan L· You]. Theorem 1 If α and b are η-tuples, r and s integers, 1 < r < s < n, then ^,y"j^LY'J^L)"\ (I) Л \& + b)J \е!ГГ1(й)/ \е1ГГ1(Ь) with equality if and only if either a~b, or r = s = 1. D First assume that r=l, when we can clearly also assume that 2 < s < n, and that a and b are not proportional. We give two proofs of this case.

Means and Their Inequalities 339 e[n] (a) (i) Write gs(a) = 'n ~ , фа{а,Ь) = gs(a + b) - gs{a) - gs(b), and as usual e[n \a) An = Σ?=ι ai, Bn = £?=1 bi. If s = 2 then 02 (а, Ь) = ^T1 г ™ г n > 0, which completes the proof in 2AnBn(An + Bn) this case. Now assume that s > 2. The following identities are easily demonstrated. (a) eel?1 (fl) = Σ^«if-i1 &): (/?) ejf]Ы = «Д"" &) + еЦ^); (7) (η - 8)eW(a) = £<£!,&); W s45](«) = А>екв1(й) - Eafe^1^). From these we can deduce the following: 1 n / 2 ·(α,β=«Μα4+Λ-ι(9ί) + (<Ч + bi)2 bi+9s~i(ki) ai+bi+gs-^ + bi) (3) We are now in a position to complete the proof of this case, by induction on s. By the induction hypothesis g 3-1(04 4-έί) > <7s-i(&0 + <7s-i(I^)> with equality if and only if д/{ is proportional to b^. Then using (3), and the induction hypothesis, <t>s(a,b) (4) * ^ \(oi +pe-i(ai))(bi +pe-i(^))(ai + bi +pe-i(a<) + £s-i(^))/ Further if for at least one г, а^ is not proportional to b^ the first inequality in this expression is strict and so φ3(α, b) > 0. If on the other hand for all г there is a λ^ > 0 such that д/{ = \У{ then (агд3-Ж) ~ bigs-Mi)? = (α* - λΛ)2^^), which is positive since a is not proportional to b. So again ф8(а,Ъ) > 0. and the proof of this case, r=l,l<s<n, is completed. (гг) The inequality is immediate in this case from I 4.6 (~19) if we can show that gs(a) is concave. This we show by induction on s and note that it is immediate if s = 1. Now suppose gs-i(a) is concave and consider (2). The function x2/(x-\- y), x, у e R+, is convex by I 4.6 Theorem 40 (g), increasing in χ and decreasing in y. So, by (2), I 4.6 Theorem 40 (b) and (d), gs(a) is concave; [Soloviov].

340 Chapter V Now suppose that r > 1. Then by (1) in the case r = 1, eks~rl(a + b)/ Ш eks-jl(a + b) l/r ^=V еГ,](й) eTJJ(6) Then using (H) in the form III 2.1(4), eV(a) \1,Г+{ βίίϋΟ Ч ^ The cases of equality are immediate from those for r = 1 and (H). D Remark (i) Inequality (1) can be interpreted as saying that the function gSir(a) : α ι-» , n_ ~ is concave; the case r = 1 was proved directly in proof (гг) above. Remark (ii) In particular (1) shows that gSir(a) is an increasing function of ai for each г, 1 < г < η. This however is easily proved directly. Using the identity (β) above, and 2 Corollary 2(a), дд3,Л&) = eif~r](a)ej1'I1ll(g;)-ei;1(a)ei;i1,-l](^) ^ (ek^fe))2 = etr'KR5:^) - ejfl^jelfir11^) 0 (вГг](а))2 Corollary 2 if α and Ь are η-tuples, r and integer, 1 < r < n, then ekrl(fl + b)>6lrl(«) + ©kr1®- (5) with equality if and only ifr = 1, or a~ b. D (г) This is just the case s = r of Theorem 1. (гг) Alternatively we can use the case r = 1 and the fact that the function gs(a) is concave, see proof (гг) in Theorem 1. Clearly (sli (a)J — ( Щ=10г(^)) · So

Means and Their Inequalities 341 by I 4.6 Theorem 40(e) Un\a)) is concave and the result is just I 4.6 (~19); [Soloviov]. D Remark (iii) Proof (г) in Theorem 1, and Corollary 2, is that in [Bullen &; Marcus; Marcus L· Lopes}; the proof in [McLeod] is entirely different. Proof (гг) is due to Soloviov. 5 Complete Symmetric Polynomial Means; Whiteley Means 5.1 The Complete Symmetric Polynomial Means The definition of elr](a) can be rewritten as: 4ρ](β) = Σ(ΙΝ)' d) where the sum is taken over all (™) η-tuples (гх,..., in) with ij = 0 or 1, 1 < j < n, and Σ"=ι 4 = r· A natural generalization is the rth complete symmetric polynomial of a4, ч% {о), (n -f- r — 1\ j non-negative integer n-tuples (г15... ,гп) with Σ™=1 ij = r. In addition we define 4J(a) = 1; [AI pp. 105-106; Dip.55; MPFp.165]. Example (i) It is easily checked that: 4г1] (α) = e£](α); c[? (α) = a\ -\ h a^ + αια2 Η l· an_ian; [3l / \ Ч Ч Ч 9 9 9 9 9 9 сз (&) = a1 -{- a2 -l· a3 -{- oxo^ 4- а^з 4- o2as 4- &2αι 4- t^ai 4- α3θ2 4- 0,10,20,3 Associated with the rth complete symmetric polynomial is the rth complete symmetric polynomial mean5 : 1/r (3) ОМ(а) = (qM(a))1^ = ((U%) The complete symmetric polynomials can also be defined in a manner analogous to 1(8): Πα -α,*)-1 = Σ44ω** = Σ ( к" kw. (2) г—ι к—о к=о ^ ' Lemma 1 If αϊ,..., αη are n distinct real numbers then 4rl(s) = fe^n+r-1]n-1. (4) Also called complete symmetric function; see Footnote 2. Also called the complete symmetric mean; see Footnote 3.

342 Chapter V D If the rational function on the left-hand side of (2) is written in terms of its partial fractions i —ι i—ι then evaluation of the coefficients Д· leads to (4); see I 4.7 Example (ii) and [Milne- Thomson pp. 7-8]. D Corollary 2 If к is even and if for у е R, Y*™0 i*4"^"1)^ > 0 where b2rn > 0, then J2i^oCn (а)ЬгУг > 0 provided α is not constant. If к is odd the same result holds provided α is non-negative and not constant D From Lemma 1 if ПУ) = Έ'Ζ <%+i](fi)W and f(x) = xn+k-i Σ·Γ„ biixyf then F(y) = [a; /]n-i, and so to say that F is positive is equivalent to saying that / is strictly (n — l)-convex, see I 4.7. Now /(η-1}(ζ) = (η - 1)}.хкТ,2г=о С^п-^ЩхуУ which is non-negative under the above hypotheses. Since / is a polynomial of degree at least η we have from I 4.7 Lemma 45(d) that / is strictly (n — l)-convex, which completes the proof. D Remark (i) This is a result of Popoviciu and is the basis of the first proof of the next result that is analogous to S(r;s); [Popoviciu 1934a, 1972]. Theorem 3 If α is an η-tuple, r and s integers, with 1 < r < s then й[:](а)<й)°\а), (5) with equality if and only if α is constant. D Using the method of proof (г) of 2 Theorem 3(b) it is sufficient to prove that (я1г1)2<я!гЧг+11, (6) with equality if and only if α is constant; this is an analogue of 2(1). We give two proofs of (6). (0 Put m = 1 and к = (?)/(*^-1) in Corollary 2 then ^Lo {к+п-1")Ь^ = (1 + у)2 > 0. So by that result Σc["+i]teW = № + ^[n+1] + уЧ+2] > о, with equality if and only if α is constant. Putting к = r — 1 implies (6).

Means and Their Inequalities 343 (ii) Let A = {(xlr .. ,xn-i); Жг > 0, 1 < г < η — Ι,Χι Η hxn-i < 1} and Жтт, —- ι Χι " ^n—1· J-nen ЦпЧа) = (ri - 1)\ / (a1x1-\ hanxn)rdxi .. .dxn-i, and (6) follows by an application of (C)-J, see VI 1.2.1 Theorem 3(b). D Remark (ii) This second proof is due to Schur; see [HLP p. 164]. Remark (iii) A completely different proof has been given by Neuman, who has given the following interesting generalization of (6): cfc+v]<{№])llV{<\tv'])lh\ (7) where ρ > 1, ρ7 is the conjugate index, and u + v,up,vp' £ N*; putting ρ = 2, и = (г - l)/2, ν = (г + 1)/2 in (7) gives (6); [Neuman 1985,1986; Neuman L· Pecaric Η The above integral for q£ (a) can be used to obtain the following extension of (6); see [HLP ρ.164]. Theorem 4 И α is a real non-constant η-tuple then Σ™8=1(\η (α)χ j>xg is a strictly positive quadratic form; and if α is a positive non-constant η-tuple then Sr,s=i Яп 8+1]{а)хгх3 is strictly positive. It is not known if the following inequality analogous to 4(1) is valid for the complete symmetric polynomials: <Jn-r](a + b)J ~ \c[rr](a)J VciTrl(6) although the particular cases s = r, s = r -{- 1 have been proved by McLeod and Baston respectively; in particular this gives an analogue for 4(5), see below 5.2 Remark (iv): [Baston 1976/7; McLeod]. An inequality between the elementary symmetric polynomial means and the complete symmetric polynomial means is given below 6 Corollary 6. For further results concerning the complete symmetric polynomials see the above references and [Baston 1976,1978; DeTemple L· Robertson; Hunter D]. 5.2 The Whiteley Means Identities (2) and 1(8) suggest the following generalizations of the elementary and complete symmetric polynomials.

344 Chapter V Let α be a real η-tuple, s e Ш, s =£ 0, к e Ν*, and define the sth function of degree k, tts](a), by 1 + £>1M («)** = П^+ЫхУ Hs>0, fc=i inLf1"^)1 ife<0. The WhiteJey meaas are defined as6 l/fc (8) ί / \ 2ВкМ(й) = if s > 0, if s < 0. (9) An alternative definition of tn (a) is: £n ' (a) = Σ (Π£=ι -4/ajJ') > where if s > 0, λ* = < (-1)г if s < 0, (10) and the summation is over all non-negative integral η-tuples (ii,... in) such that Remark (i) By analogy with Sn and Qn^ we will write wln'8] for (W[^s])k. Example (i) If s = 1, and λο = Ι,λ^ = 0 otherwise, then from (10) we have tn' = en , this is also immediate from (8). [fc,-i] J*l. Example (ii) If s = — 1 then λ^ = 1 for all г, and so in J = Cn1; this is also immediate from (8). Example (iii) Simple calculations show that W^s\a) = 21п(й)> and tt»l; 1 η η ^π((8-1)Σ°«? + 2βΣα*αί) n(ns — 1) Remark (ii) If s < 0 then the coefficients in tn 'sj are all positive; this is also the case if s > 0 and either 0</c<s-f-l,s not an integer, or s is an integer and 0 < к < ns. This explains certain restrictions in the theorems given below. Note that if s<0 then (-l)fc (nfcs) = (~nai'fc~1).

Means and Their Inequalities 345 We note that an expression for rn ,s* in terms of an integral can be obtained; [BB p.36]. If s < 0 and \t\ is small enough then 1 f°° i1 " ai)S = 1V / e-s(1-et)x-i_1 dx. So n -I />0O ЛОО П n Π(1-α^)5 = - ττττη / ···/ exp(- Va^l - а^))Д х~5~Мх1 ■ · · dxn. »=i Ц-5-1)1; ^0 Jo i=1 i=1 Hence 4M («) = -ι /»oo /»oo n / n \ n щ=гл)1г Уо ■' л(£ ад) ехрг £Ж1 νδxjs "ldri-da!-(ii) In this section we extend various properties of the elementary and completely symmetric polynomial means to the Whiteley means and in 5.3, by considering even more general means, other properties will be extended. First we note that the Whiteley means have the property of strict internality. Lemma 5 If a is an η-tuple, к an integer, 1 < к <n,sGl,s^0, then mina < W^,s\a) < max a, with equality if and only if α is constant. D This is immediate from (8) and (9). D Before generalizing other properties of the elementary and completely symmetric means we establish some lemmas of Whiteley; see [Whiteley 1962b]. Lemma 6 If 1 < г < n, then Зщ +аг дсц ~Stn {-h S>U' Ц^.^е^Ш.^^^ .<а D If s < 0 we have from (8): oo гч [fc,s]/ ч η σο (1-αίΧ)Σ да хк = -зхЦ(1-акхУ = -8Х^ф^(а)хк. к=0 г fc=l k=0 This gives the result in this case. The proof for s > 0 is similar. D

346 Chapter V Corollary 7 D Sum the results of Lemma 6 over г and the lemma follows easily from Euler's theorem on homogeneous functions, see I 4.6 Remark(ii). D Lemma 8 If s < 0 then 2nlM1(a)<2uL2,sl(fi)· If s > 1 this inequality is reversed and in both cases there is equality if and only if α is constant. D Prom Example (iii) η2((20ίϊ··1(α))2 - (W^(a))A =^((» - 1)£>? -2$>a,·) ^ ' i—\ i<j ns-1 ^V JJ i<3 from which the result is immediate. D The next result, the main result of this section, generalizes both S(r;s) and 5.1 Theorem 3. Theorem 9 If a is a non-negative η-tuple, and s > Q, and if k,m are integers with l<k<m<s-\-l when s is not an integer, and 1 < к < m < ns if s is an integer and then m[™>s\a)<m[^s]{a). (12) If s < 0, (~12) holds. In both cases there is equality if α is constant. It follows from the proof of 5.1 Theorem 3, and of proof (i) of 2 Theorem 3(b) that it is sufficient to prove the following generalization of 2 (1) and 5.1(6); [Whiteley 1962 a). Theorem 10 If α is a non-negative η-tuple, and s > 0, and if к an integer with 1 < к < s when s is not an integer, and 1 < к < ns if s is an integer and then Kfc'sl(a))2 > ^-^(a^+^ta). (13) If s < 0, (~13) holds. In both cases there is equality if α is constant. D If η = 1 the result is immediate for all /с; if к = 1 the result reduces to Lemma 8. The proof is completed by a double induction on к and n.

Means and Their Inequalities 347 Assume the result is known for all к and all n', n' < n, and all η and all к', к' < к and put A = {a; a is a non-negative η-tuple and ton ~ ' (a) ton ' (a) = 1}· Under the given hypotheses A is compact and so, on A, (ton (a)) must attain its minimum if s > 0, its maximum, if s < 0. In either case call this value M. By Euler's theorem on homogeneous functions, see I 4.6 Remark(ii), ^.(■Г-у^) . аюГ1,1(й)т,*«..1(й). 2k, so that in A not all the partial derivatives of ton (a)ton ' (a) vanish together. Let us first assume, that Μ is attained at a positive η-tuple, a0 in A say. Apply Lagrange multiplier conditions, see II 2.4.3 Footnote 10, at this point: 0(юкм(о0))2 x0(tokfc-M(a>k*+M](fi0)) П1^.^ ^° λ ад = o,i<*<n, α [fe,s] r\ [k — l,s] r\ [fe+l,s] Taking each of the identities in (14), multiplying by сц, adding and using Euler's theorem on homogeneous functions gives λ = Μ. We now obtain an upper bound for λ. Add the identities in (14) and use Corollary 7 to obtain 2kto^to^-1·^ = A((fc - l)roilfc+1's]Wlf_2's] + (*: - 1)№{?"М1юкМ)' which on simplifying gives [k + l,e] lk-2,,] 2к-\{к + 1) = \{к-1)П» 7" ■ (15) Now as λ = Μ = (№kM])2Mfc"Mtokfc+Ml, puttingм = (»к*"1,в])2/(»5.*"2'в1»п,л1) (15) becomes 2fc-A(fc + l) = ^—^. (16) The inductive hypothesis is μ < 1, if s > 0, and μ > 1 if s < 0; substituting in (16) leads to λ < 1, if s > 0, and λ > 1 if s < 0 which completes the induction. In addition if λ = 1 then μ = 1 and the induction applies to the case of equality as well.

348 Chapter V Now consider the case when Μ is attained at point with only ρ non-zero coordinates, ρ < n. This reduces to the previous case with η replaced by ρ and the proof is complete. D Remark (iii) From Examples (i) and (ii) it is immediate that (13) is a generalization of 5.1 (6) and of 2(1). However here we need a to be non-negative, see the comment in proof (iv) of 2 Theorem 1. Example (iv) Consider for instance s = — 1, η = 2, к = 2, a\ = —α<ι = 1. Then tt42'~1](a, -a) = a2 but m^'~1](a, -a) = ΐυί>3'~1](α, -a) = 0. COROLLARY 11 If s > 0 and if к an integer with 1 < к < s when s is not an integer, and 1 < к < ns if s is an integer and if α is a non-negative η-tuple then (ίί?·ί1(α))2>(4*-1·ί](α))(4*+Μ(«)); the reverse inequality holds if s < 0. D This an immediate consequence of Theorem 10, and the observation that ns V ( > 1 if s>0, fc-1 ns \ / ns k + l)\k I < 1 ifs<0. D Corollary 12 If s > 0 and ifk, m integers with 1 < к < m < s when s is not an integer, and 1 < к < m < ns if s is an integer and if α is a non-negative n-tuple then the reverse inequality holds if s < 0. D This follows from Corollary 11 just as 2(6) follows from 2(1). D Whiteley has also extended 4 Corollary 2; [Whiteley 1962a]. Theorem 13 If s > 0 and if к is an integer, к < s 4-1 if s is not an integer, then for non-negative η-tuples a,b, W[^s\a + b)>wts\a)-l·W[^s](b). (17)

Means and Their Inequalities 349 Ifs<0(~18) holds, and the inequalities are strict unless к = 1 or α ~ b. Π (г) Inequality (17) is equivalent to an analogous one in terms of tn (a) which follows from (11), using a form of (M)-/, see VI 1.2.1 Theorem 3(c). (гг) A direct proof following a method similar to that used to prove Theorem 10 is given in [Whiteley 1962a). However since the three cases, s negative, s positive and not an integer, and s a positive integer, need separate treatments the proof is rather long. A neater proof of a more general result is given in the next section.D Remark (iv) If s = 1 then (17) reduces to 4(5), while if s = —1 we get Mcleod's analogous result for the complete symmetric polynomial means: [McLeod]. U[:](a + b)<Q^(o)+Q^(b). Remark (v) If s = —δ, where δ is a small positive number then tn'~ (α) = ΣΓ=ι ai + °^2)- APP!ying (17) in this case and letting δ —»■ 0 we see that (17) implies (M). Remark (vi) Inequality (17) is equivalent to saying that the surface 2Π^'^(α) = 1 in the "positive quadrant " of Rn is convex if s > 0, and concave if s < 0. Remark (vii) Other results can be obtained by noting that (13) says that №п is a log-convex function of к . 5.3 Some Forms of Whiteley Let a^·, 1 < г < η, г = 1,2,..., be η sequences of positive numbers and define inductively the η sequences of positive numbers 1 r Pi,j, 1 < г < п,г= 1,2,..., by a^r = — JjAj- ' j=1 If α is a non-negative η-tuple, θ > 0 define the functions ghJ (a) of degree к by OO П / OO \ПуОО Σ^1(^* = <,Π(1 + Σα«.··(β*:Β)Ί = ίΙΙ(1+Σ( k=0 г=1 ^ r=i ' г=1 ^ r=i llj=lPi,j \, чГ -^Ti—)(a^) (18) [Bullen 1975; Whiteley 1962b]. Example (i) The vn of the previous section are particular cases of gn ; in (18) take Pi j = s-j'4-lifs>0 and Pij = -s 4- j - 1 if s < 0. Example (ii) We can generalize the previous example by letting σ be a positive or negative η-tuple, and taking β^ — σ^ — j 4- 1 if σ > 0 and β^ = —σι 4- j — 1 if σ < 0. In this case g\i ^ would be written t\i'~'. Clearly if o_ is constant, s say,

350 Chapter V then tn — tnS. These particular functions of degree к can of course be defined directly by writing: ~ r, , ΓΠ?=1(1+^)σ<, *<Ι>0, fc=i (ilLt1-^' ίίσ<0. The functions tn and their associated means were introduced by Gini, and studied for statistical purposes by various authors, who in addition used these functions to define various biplanar means; see 7.3. However because the main interest was in finding suitable statistical means these authors seemed to have introduced more means than they proved theorems about them; [Gini 1926; Gini Sz Zappa; Pietra; Pizzetti 1939; Zappa 1939,1940]. Later these functions were studied by Menon, who obtained most of their known properties; in particular he extended 5.2 Theorems 9 and 10 to this more general situation; [Menon 1969c,1970]. Certain restrictions have to be placed on к to ensure that the coefficients in tn are positive; this is similar to the case see 5.2 Remark (ii). If σ < 0 then no restrictions are needed, all к are permitted. When σ > 0, and miner is not an integer then we require that 1 < к < 1 + miner. If miner is an integer then the restrictions are more complicated to state, and for this reason this case has usually been excluded from consideration; [Menon 1970,1971]. However the proofs only require the coefficients mentioned above to be positive and so extend to this case when it is known that the coefficients are in fact positive. Thus if σ is constant, s say, then it is sufficient to have 1 < к < η, as we have seen. Example (iii) Menon has made further extensions of the previous examples; [Menon 1971]. If 0 < q < 1 then the q-binomial coefficients are defined by7: ί Λ 1 - qs-i+1 11 l-o* ' 1, l>0. if к > 0, if к = 0, if к < 0. If then σ is a positive η-tuple define e„ (q;a), and c„ (q;a) as g„'(a) with a,r given by ι and г respectively. In the above reference Menon has Lr J L r J extended Theorem 10 to this situation. 7 Note that limq_+i [£] = (£)·

Means and Their Inequalities 351 Example (iv) Menon also used the ^-binomial coefficients to generalize Sn (a) as follows: η jr Clearly: s[n\ (a) = sn(a); *n,]o (a) = e[i] (a). Is [k Using this Menon proves As we see in the proof of Corollary 15 below is log-concave as a function of k. 5i,V](a)<il](a)<(^,(fi))2, an inequality containing both 2(1) and 2(2) as special cases; [Hon; Menon 1972]. The original proof of 5.2 Theorem 13, [Whitely 1958], used 5.2 Corollary 11, or 5.2 Theorem 9, in a fundamental manner. Such results are not available in the present more general situation. However as Whiteley pointed out slightly weaker results will suffice; [Whiteley 1962b]. Let us consider the case s > 0; then inequalities (13) and (12) imply the following weaker and simpler inequalities, with the restrictions on к and m given in Theorems 9 and 10; (4M)2 > (nr) 'if-1''1 4?+1'e]; (*!4M)1/fc > (m!*M1/m- (19) If s < 0 then the inequalities (~19) hold. It is these results that will be extended, and which will then be used to obtain Theorem 16 below, a theorem that contains 5.2 Theorem 13 as a special case. If s > 0 then inequalities (19) imply the following still simpler and easier inequalities: (ί[Μ}2 > ί{ί-Μ] 4*+Μ. (ί{Μ)ΐ/* > (ίΚ.1)1/-. (20) However if s < 0 we cannot deduce (~20); in fact as we will see below, (24), the same inequalities, (20), are valid if s < — 1. Theorem 14 (a) If α is a non-negative η-tuple and if in (18) for each г, 1 < г < η, the sequences cxiT, r = 1,2,.. 1 < i < n, j = 1, 2,..., then j - 1 the sequences щг, r = 1, 2,... are log-concave, or equivalently Pij-i > —:—Pij, (fllf1) ^ 9[tl] fliT11, к > 1; (flW)1^ > (ffiT1) /TO, 1 < Л < m. (21)

352 Chapter V (b) If instead for each i, 1 < i < n, the sequences air, r = 1, 2,... are strongly log-concave, or equivalently Pij-ι > Pij, 1 < г < ?г, j = 1, 2,.. ., then (<a2 > (^±I) 9[tA ffjf+l1, к > 1; (9W)1/k > (5H)1M 1 < к < т. (22) If the hypothesis is changed to weakly log-convex then inequalities (~22) hold. D (a) In the case of positive sequences the left-hand inequality in (21) follows by a simple induction from I 3.2 Theorem 8(a). If some of the αι are zero then consideration of (18) shows that the inequality follows from the inequality for smaller values of n. The right-hand inequality in (21) follows from the left-hand inequality using the arguments of 2 Theorem 3. (b) This has the same proof as (a) except that now we use I 3.2 Theorem 6. D Using the Remarks (iv) and (v) that follow I 3.2 Theorems 6 and 8 we can make the following observations about the cases of equality in the above result. (i) The left-hand inequality in (21) is strict unless a is zero, or if a^ = 0, г =fi j, and a2jk = aj^-iOij^+i· (ii) The right-hand inequality in (21) is strict unless a is zero, or if сц = 0,г ^ j, and α| = щ^-гЩ^+г, 1 < s < к. (iii) The left-hand inequality in (22) is strict unless a is zero, or if a? · = a^j-^ij+i, 1 < i < n, 1 < j < к 4-1. (iv) The right-hand inequality in (22) is strict unless a is zero, or if щ = 0,г φ j and а2^к = ajjk-iOSj,k+i' Corollary 15 If a is a positive η-tuple, σ a real n-tuple, 0 < q < 1, к a positive integer then: (4^(α))2 > *±1(ίί*-^(α)) (t[*+1^(o)), σ > 0; (23) if σ < 0 then f~ 23) holds; (4M(a))2 > (*к*-1523(о))(4*+1:а1(а)), 2 < "I; (24) (ekfc;d(9;a))2 > (е^(«;а))(е£+1^(д;а)), σ > 0; (25) (clf;2](9;a))2 > {&-^{q;a)){£+l^(.q-a)), σ > 1. (26) D From Example (i) it is easily seen that Pij satisfies the condition in Theorem 14(b) if σ > 0, while if σ < —1 it satisfies the condition in Theorem 14(a). This

Means and Their Inequalities 353 by Theorem 14 implies inequalities (23) and (24); that they are strict follows from the above discussion of the cases of equality. Similarly using the definitions given above inequalities (25) and (26) follow from and s 4- r — 1 r are log-concave Theorem 14(a) if we show that the sequences when s > 0. We will only consider the first case, tho other can be handled in a similar manner. Since [б s 1 _r — lj Ί 2 Г s [r + l_ (1 r.S — Г+1 )/(l-9r) (l-9-r)/(l-9r+i)' it suffices to show that f(x) = (1 — qs ж+1)/(1 — qx) is decreasing, 1 < χ < s 4- 1. Now f{x) = bg^ (<f ~ж+1(1 - дж) 4-g*(l - qs~x+1))> which is negative since 0<q<l. D Remark (i) Inequality (24) justifies the remarks made above about (20). Of course inequalities stronger than (23) are known when σ is constant. All the results in Corollary 15 are due to Menon, who also obtained similar results for a type of function not being considered here that also generalizes rn ; [Menon 1969b,1970,1971]. In particular (25) implies the inequality of Menon quoted in Example (iv). We are now in a position to prove a generalization of 5.2 Theorem 13. Theorem 16 If a and b are non-negative η-tuples, and if the sequences a^r, ,, 1 < i < n, are all strictly strongly log-concave, or equivalently if Pij-i > ftj, 1 <i <n, j = 1,2, (27) then \i/fc l/fc Л/к (№(а + Ъ)ГК > (9Ш< + (<Л)Г/К, к > 1 (28) If the hypothesis is changed to weakly log-convex then (~27) holds. In both cases there is equality if and only if к = 1 or а~ b Before starting the proof we make some preliminary remarks. Remark (ii) The above discussion shows 5.2 Theorem 13 is a particular case of this result. As a result, the following proof completes the discussion of that theorem. The method of proof is similar to that used in 5.2 Theorem 10, as stated in the discussion of that theorem. However Theorem 14 allows us to make a subtler

354 Chapter V use of the Lagrange multiplier conditions. Further the use here of the more general class of functions of degree /с, к £ Ν, allows us to differentiate and stay within this class. As is easily seen, if gn is a function of degree /с, к > 1, then ддп /дат is a function of degree к — 1; further if the coefficients of gn satisfy the above hypotheses so do those of the partial derivatives ддп /даш, 1 < m < n. Remark (Hi) It is easy to show that if a = (g/, α"), gf = (αϊ,... ,am), a" = (am+i,... , an) then <#](<0 = ^>№)яК(й")· r=0 D The proof is by induction on k. If к = 1 there is nothing to prove as (28) is trivial in this case. However to consider the cases of equality the induction must start from к = 2. In that case squaring (28) twice leads to the equivalent inequality 5^?,j=i /?i,i/?j,i(A,2/?j,2~A,ift',i)(^bj— djbi)2 < 0, which is an immediate г > j consequence of (27). Now put A = {(a,b); a > 0,b > 0, and gn (a + b) = 1}. This set is a compact subset in R2n. So the function (gn(a)) 4- (<7n (έ)) will attain its maximum on A if (27) holds, and its minimum if (~27) holds. In either case call this value M. It suffices, from considerations of homogeneity to prove in the first case that Μ < 1, and in the second case that Μ > 1. Since both cases proceed similarly let us consider the first case only. Suppose that Μ is attained at a point (a, b) of A satisfying a{ > 0, i = zi,..., гП1, and di = 0 otherwise; bj > 0, j = ji,...,zn2, and bj = 0 otherwise. We can assume that 1 < щ, ri2 < n, since the cases when either, or both, of щ, η<ι is zero are trivial. The proof breaks into two cases: (г): for some a, aqbq > 0; (гг): for all г, aibi = 0; then in this case (zi,... ,гП1), (ji,... ,гП2) are distinct subsets of (1,..., n); put a' = (a^,... ai ), У — (bj1, · ■ ■ bj ). Proof of case(z). Apply the Lagrange multiplier conditions, see II 2.4.3 Footnote 11, to the non-zero variables сц, bj and proceed as in 5.2 Theorem 10 to obtain W1(B)1/"-AJ|:(9i,'l(a + S)1", = o,j,,„...,fcI. βο:0№))'" - а£-Ы,"(« + й)"' = о. * = *..■··,<»„ l-W'iB)"*-^ l/j ^I7j

Means and Their Inequalities 355 In other words, (gW(a)f^/k Щ^ = X(9W(a + b)f-k)/k Ц£±М (29) mk)f-^ &. = x{gw{u + b)f-k»k d-^^. (30) Multiply (29) by сц, and (30) by bj, and add, г = zi,..., ini j = ji,..., jn2, and use Euler's theorem on homogeneous functions, see I 4.6 Remark(ii), to get (9[kW/k + (№(k))1/k = Ч9[кЧа + Ъ))1/к. (31) Now take i = j = q in (29) and (30), raise both to the power l/(k — 1), and use (31) to get /dffnfe](a)xi/(fc-i) + /dffn](fr)xi/(fc-i) = Xk/(k-i) гд9п\а + k) ,i/(k-i) ^ daq ' ^ dbq ' ^ daq ' dg[n] (a) By Remark (ii) —^—— is a function of degree (k — 1), satisfying (27). So the OCLq inductive hypothesis tells us that λ < 1; but by (31) Μ = λ, so Μ < 1 as had to be proved. Proof of case(zz) . Using the notation introduced above (28) reduces to (ffJfV+tf))1" > (9[кЧа'))1/к + (9[k](b'))1/k- or, using Remark (iii), £*№)$<*')) * №'))1A + Ш))1/к- Raising this to the /cth power leads to the following inequality that has to be proved. E№)s£-p]oo ± Σ (f)(9[k}(a'))r/k(9[km{k'r)/k- (33) r=0 r=0 ^ ' If 1 < r < /c — lwe have by the right-hand inequality (22) that gn\(g!) > V yniK-}) , and g[k2 r](b') > (l~ V * ^equality (33) is then an immediate consequence of these inequalities. D

356 Chapter V Remark (iv) By applying Theorem 16 to the functions tn Theorem 12 can be extended to such functions, as suggested in [Whiteley 1958, p.50]. 5.4 Elementary Symmetric Polynomial Means as Mixed Means Elementary symmetric polynomial means are particular cases of the mixed means introduced in III 5.3: Θ^(α) = Шп(г,0',г;а). So we could compare, using a matrix analogous to that in III 5.3, particular cases of III 5.3(12) with S(r;s) as follows. /ЯЯп(0,1;1;а) ЯЯп(0,1; 2; а) ... ЯЯп(0,1; η - 1; а) ЯЯп(0,1; η; α) \ ЯЯп(1,0;п;а) ЯЯП(1, 0;п - 1;а) ... ЯЯП(1, 0; 2; а) ЯЯП(1, 0; 1; а) . V ©1П]Ы б1п-1]Ы ··. б1?](а) ©11](а)) У Then by S(r;s) and III 5.3(12) the rows increase strictly to the right, and the entries in the second row are strictly less than those in the first row, and strictly greater than those in the last row, except for the first column all of whose entries are 0n(a), and for the last column all of whose entries are 2ln(a) . An immediate problem suggests itself— are there any further relations between the elementary symmetric means in the last line and the mixed arithmetic and geometric means in the first? Carlson, Meany & Nelson conjectured that if r-fra > η then SWn(r,0;r;a)<SDtn(0,l;m;a); (34) [Carlson, Meany L· Nelson 1971b). If this were proved to be correct the above matrix could be rewritten /ЯЯп(1,0;п;а) ЯЯП(1, 0;n - 1;а) ... 9ЯП(1, 0; 2; а) 9ЯП(1, 0; 1; а) \ ©Ιη](α) 6[Г1]Ы ··· ©12]Ы ©11](а)) . \ЯЯп(0,1;1;а) 9ЯП(0,1; 2; а) ... 9ЯП(0,1; η - 1;а) 9ЯП(0,1; п; о) / Now the rows again increase strictly to the right, but the columns strictly increase, except of course the first and last. The particular case η = 3 of (34), 9Я3(2,0; 2; а, Ь, с) < 9Я3(0,1; 2; а, Ь, с), was proved in [Carison 1970a]. It is the following lemma that should be compared with III 5.3(17). Lemma 17 If a,b and с are positive numbers then аЪ + Ъс+ са 3 (a + b)(b + c)(c + a) .

Means and Their Inequalities 357 D (a 4- Ь)(Ь 4- с)(с 4- a) =(a 4- Ь 4- c)(ab 4- be 4- ca) — abc = (a + b + c){ab + bc+ca)- ^ИХИО) ^УаТс о >-(a 4-6 4- c)(ab + bc + ca), by (GA), 3/2 ab + bc + ca V 3 ' byS(r;s); and this is just (35). D More generally Kuczma has shown that: Жп(2, 0; 2; a) < 9Jln(0, l;n - 1; a), and ЯЯп(п - 1,0; 2; a) < Ш1П(0,1; η - l;a). Both of these inequalities include (35); [Kuczma 1994). 6 Muirhead Means A very different generalization of the elementary symmetric polynomial means is given as follows. Let α be a non-negative η-tuple, and write \a\ = a± 4- · · · 4- ctn\ then if a is an η-tuple we write Π j=l and define the Muirhead α-mean or just Muirhead mean of α as, 21η,α(α) = 2tn,(aiJ...>an)(a) = (nXnfea))1 ~; (2) of course if \g\ = 1 then 21п,а(й) = mn (&;«)· Remark (i) Clearly these means have the properties of (Co), (Ho), (Mo), (Re), (Sy), and are strictly internal. Convention Since the order of the elements of in α is immaterial we will always take them to be decreasing. , α<*\)β _|_ αβ})θί χ 1/α-\-β Example (i) The following are easily checked. 2(.2,«,/?(&, b) = ( ) ; 2tn(a) = 2in,(i,o, . ,o)(a); 0n(a) = 2ln,(i/n, i/n,...,i/n)(a) = Stn,(i,...,i)(a); r terms ая{Г](й) = Я„,(г,о,..,о)(й), Γ?έ0; 6M(fi)=2l«,a(a), a = (ί~^7ΐ,0,... ,0). Example (ii) Using 5.1 Example(i) we can readily see that 431 («) =^1(аз,(1,1,1)(й).5»з,(2,1,о) (а), Яз,(з,о,о) (fi); 1,6,3)) The main purpose of this section is to obtain conditions under which two different Muirhead means are comparable. The answer, Muirhead's theorem, is in terms of the order relation defined in I 3.3.

358 Chapter V Theorem 1 [Muirhead's Theorem] Leta^, anda^ be non-identical non-negative η-tuples, α an η-tuple; then mn(a; a^) and mn(a; a^) are comparable if and only if one ofg?1' or a^ is an average of the other. More precisely mn(a;a(1)) < mn(a;a(2)) 4=^ a(1) -< a(2\ (3) or, a^ is an average of ο№\ Further the inequality in (3) is then strict unless а is constant. D (a) => Suppose that the inequality in (3) holds for all positive η-tuples a. Choose a constant and equal to a, when the inequality becomes a'— I < a'- L By considering large a and small a this implies that \a^\ = \а_(2Ц, that is the first part of 13.3(16). Now take a\ = · · · = a^ = α, α^+ι = -·· = αη = 1,1<&<η. Then the largest power of a in mn(a; c^1)) is J2i=1 ot\ , whereas in mn(a; c^2)) it is Y^i=1 oq '. If we now assume that a is large then the inequality implies the second part of I 3.3(16). Thuso^1) -<a{2). (b) <=. We give two proofs of this implication. (г) It is sufficient to note that φ(α) = ττιη(α;α) is both symmetric and convex and apply 1 4.8 Theorem 49. (гг) By I 3.3 Lemma 13, Muirhead's lemma, it is sufficient to consider the case when o^1) = o№T and Τ — λ/4- (1 — λ)<2, where Q is a permutation matrix that differs from / in just two rows. Without loss of generality we can assume that 41} = Xa^ 4- (1 - λ)<42), <41} = (1 - \)a?] 4- λ^2), a£] = ctf}, 3 < к < п. Then 2(η!)(τπη(α;α^)-τπη(α;α(1))) N^.TT "?\ ^ί2) ^2) , «ί2) «i2) <*P «P «21} «?\ i=3 ^,ΤΤ "j0/ ^^/ «i2)-^2) , «i2)-42) J=3 _ λ(«[2,-«?,)η(1-λ)(α(1ί,-α?)) _ (1-X)(a^-a^) X(a^-a^U aii aii aii aii ) IX Σ!Πβ«; («i^j0* (4(ι 2)-4(ι 2>)= j=3 / (1-λ)(α<2>-α<2>) _a(l-A)(a(2»-a<2))s > Q The cases of equality are immediate. D

Means and Their Inequalities 359 Remark (ii) The second proof is essentially that in [Muirhead 1902/03]; see also [AIp.167; DIpp.183-184; HLP рр.Ц-51; МО pp.87,110 111; PPT pp.361-364]. Remark (Hi) Since (1/n, 1/n,..., 1/n) -<; (1,0,..., 0) we see from Example (i) that the above theorem gives another proof of (GA). It is worth giving the details. Note first that (1/n,..., 1/n) = (1, 0,..., 0)-J, where J is the η χ η matrix all of whose entries are equal to 1. Also, see I 4.3 Lemma 13, ~J = Πλ=ι ^ь ί1"-1 ° ° \ 1/1 n-k\ Tk= 0 К 0 where K= - ( , , , Kfc<n-1. V 0 0 J^k_J n-k + Лп-к I J ~ ~ Here Ij is the j χ j unit matrix or is absent if j = 0. к terms More directly let a^ be the η-tuple (n - k, 1,..., 1, 0,..., 0, 0), 0 < к < п. Then n-2 Μαη)-0η(α") = Ε Μ«;α^) -mn(a;a^+1))) A;=0 fc=0 >0; where if /c = 0 there are no factors in ^! beyond the brackets; see [HLP p.50]. Corollary 2 (a) If α is an η-tuple and if a^ and aS2^ are distinct non-negative η-tuples with a^ -< a^ then ^η,α(υ(α)<21η^(2)(α), (4) with equality if and only if α is constant. (b) [Schur] If α is a positive η-tuple and if α a non-negative η-tuple with |a| = 1 then ^n(a)<2tn,a(a)<2in(a), (5) with equality if and only α is constant, or, on the left if 21п,а(й) = ^n(&), on the right if 2ln,a(a) = 2ln(a)· D (a) This is an immediate consequence of Theorem 1 and the definition of a Muirhead mean. (b) We give three proofs. (г) By I 3.3 Example(ii) if |a| = 1 then (1/n, 1/n,..., 1/n) -< а -< (1, 0,..., 0), and the result follows from (a) and Example (i); [HLP p.50].

360 Chapter V (ii) A direct proof of (5) can be found in [Bartos L· Znam]. By (GA) we have, for each permutation (z'i,..., in) that η η Πα? ^Σαοα3· (6) Adding all these inequalities over all permutations give the right-hand inequality in (5). Again from (GA) and the definitions (1) and (2), *«*(«)> (Π! Π <01/n,) = <Ms). η j=l giving the left-hand inequality in (5). The cases of equality follow from those in (GA). (Hi) Another proof, due to Segre, can be found in VI 4.5 Example (vi); [SegreJ.D Remark (iv) Part (a) remains valid for all real η-tuples a^ and a^2\ such that aW -«a<2> and \aW\ > 0. The following interesting extension of Schur's result, (5), has been given; [Gel'man]. THEOREM 3 Let Fm be a symmetric polynomial of η variables, homogeneous of degree m, and with non-negative coefficients. Then (a) /(a) = Fmfa1/171) is concave, symmetric and homogeneous,(of degree 1); (b) д(а) = \ogoFm(e™-) is convex, (c) &^4W))1/m^ml^- (7) D (a) The concavity of / is given by III 2.1 Remark (iii), and the rest is immediate. (b) By (C), fi^M) < V7(a)/G0· Now replace α and b by e- and e-, respectively and take logarithms on both sides of the last inequality. (c) Consider first the right-hand side of (7), and put b = am. 1 n Frn(a) = f(b) =- Σ f(hi · · ■, &n, &i ■ · ■, bi-i), by the symmetry of /, i=i <f(%n(L·),..., 2in®), by the concavity of /, =On(6)/(e), ЬУ the homogeneity of /, = (Ш^(а))шFm(e).

Means and Their Inequalities 361 Now consider the left-hand side of (7) , and define b by a = exp(b/ra). Then, as above, but using (b): 1 n Fm(a) =g(b) = ~Σ f{bi,..., Ьп, Ъг,..., b;-i) i=i >^(Яп(Ь), · · · ,«n(iO) = bg (Fm(e^St-®>..., e**»®)); or, by the homogeneity of Fm, Fm(a) > Fm(e)e^® = Fw(e)(($n(a))m, which completes the proof. D Corollary 4 If 0 < ρ < 1 then mn(a; a^) < Κ(α;α(2)))Ρ 4=^ a*1* -< pa{2\ (8) D =>- This follows using the same argument as in (b) (г) of Theorem 1. 4= By Theorem 1, and (r;s) with r = p, s = 1, ™η(α;α(1)) < mn(a;pa(2)) < (mn(a; a(2)))P. D Remark (v) If we take a^1) = (r, 0,..., 0), a^2^ = (s, 0,..., 0), ρ = r/s, then the inequality (8) is just (r;s). Remark (vi) The proof of the necessity does not use the restriction on p, but Remark (v) shows that if ρ > 1 then a^ -<; pa^ is not sufficient for the inequality to hold. Corollary 5 Let a, w_ he positive η-tuples then Π(ί>»^ <Π(ί>«Λ *=* -(1)^«(2)· г—ι к—ι г—ι /с—ι D =>- This follows along the lines of Theorem 1. <^= By 1 3.3 Theorem 14 a(1) -< ap"> implies that α(1) = a^S, where S — (sij)i<i,j<n is a doubly stochastic matrix. Hence, by III 2.1 (2), (H), fc = l fc —1 j —1 fc —1 and the result follows by multiplication. D

362 Chapter V Corollary 6 If a is a positive η-tuple, r and integer, 1 < r < n, then ©Μ(α)<0Μ(α), (9) with equality if and only if either r — 1 or α is constant. D Let A = A^r\ be the set of η-tuples defined as the set {«; QL > 0, decreasing, щ = 0, r -f 1 < г < η, щ an integer, 1 < г < r, \α\ = г}. r terms The A has η elements and in particular a0 = (1,..., 1, 0,..., 0) e A. Further if a G A, a0 -< a, and so by Corollary 2(a) 2^^ (a) < 2tn,a(&)· is an rth power mean of the {%in,a}aeA, see Example (ii); and, from Example (i), 2ίη,α0(α) = β^(α); so the result is immediate by the internality of the power means, III 1(2). D Remark (vii) This result, which in the notation of 5.2 is W[^1] (a) < W[*~1] (a), was stated without proof in [Zappa 1939]. In the same paper Zappa states a much more general inequality using means derived from tn , see 5.3. This inequality reduces, in the case of constant σ, s say, to W^ ,s* (a) < W^ '~s* (a). No proof was given, although in the case of constant σ the proof of Corollary 6 can be used. The following theorem generalizes 4 Corollary 2. Theorem 7 Ifa,b, and α are positive η-tuples with 0 < |a| < 1, then %η,α(α) + 2ίη,α(έ) < %η,*{α + Ъ). (10) D First we assume that 2in,^fe) — 21η,α(έ) — 1> when by homogeneity it suffices to show that if 0 < λ < 1 then 21η,α(λα + 1 - ЛЬ) > 1. ι n ! %η,α(λα + 1 - λ b) = (- Χ]! Π(λ% + 1 - λ^Γ) Ш *(^(ΣΚλΠ^^' + α-^Π^^1)^1)^' byI112^w· > (^(Е!АП«? + (1 - *) ΓΚ/Ρ = 1, by (r;s). In general, 2tn,q(a + b) 21η,α(α) + Япй(Й 21η,α(λα' + 1-λ6')

Means and Their Inequalities 363 where a' = — r—; b = — —; λ = — , ™'~ ~ -ττ-. This shows that the argument above is sufficient. D By analogy with definition (2) Bartos L· Znam, [Bartos L· Znam], have defined, for α with \a\ = 1, a geometric mean analogue of (2): 0η>α(α)= (Π!Σ^'α^ η j=i and proved an inequality similar to (5). Theorem 8 With the above notations, ®η(α)<<δη,«(α)<21η(α). (11) D The left-hand inequality in (11) follows from (6) by multiplying all these inequalities over all permutations. Also by (GA) 1 n 0η,α(α) < — Σ}·Σ<*όα4 = Яп(а), η j=i which gives the right-hand inequality and completes the proof. D Remark (viii) In the same reference Bartos L· Znam have defined, for α with Isl = 1. 71 71—1 71 71 — 1 «ш = (ΠΣ w-i)1/n; ^(a) = ΐΣΠ^ι; j —ι г—о j=i г=о where for к > η, α^ = α^_η. Using arguments similar to those in the previous remark they have proved that &n(a) < Sl£(a) < 2in(a), and 0n(a) < 0^(a) < 2in(a). (12) However in this case some of the inequalities can become equalities even when a is not constant. Example (iii) Consider the case: сц = г, 1 < г < 3,α4 = 2, 2αχ = «2 = 2аз = «4 = ^ when <Sf"(a) = 214(a) — 2; while with the same a with 2αι = a2 = as = 2a4 = 2, 1 < г < 3, a4 = 2 we get 2if(a) = 04(a) = >/2. Example (iv) Letting a^ = г, 1 < г < η, αϊ = «2 = §,«* = 0, 3 < г < η in the first inequality in (12) leads to: 22n /2n\ ^ (2n + 2)n < < n-f-1 \ rc / (n + l) Μ

364 Chapter V Remark (ix) A further quantity similar to one in Remark (viii) has been studied in [Djokovic &;Mitrovic); see also [AIpp 284-285,3.6.46], [Mitnnovic & Djokovic], [Mijalkovic Sz Mitrovic). Remark (x) An interesting generalization of Muirhead means occurs in [Rado]. Other generalizations have been discussed; for instance we could consider: mMfea) = ^f[aV > ^Ш = i^y^fe^)) ΤΞΓ See [Beck 1977; Bekisev; Brenner 1978; Bering 1972,1973; Schonwald]. 7 Further Generalizations 7.1 The Hamy Means The following slightly different means have been considered by Hamy and others; see [Smith p.440, ex.38), [Нашу; Hara, Uchiyama L· Takahasi; Ku, Ku L· Zhang 1997; Ness 1964). Let α be an η-tuple and r an integer, 1 < r < n, then the Hamy mean of order r of α is: №а)кг)(а) = ^ЕКП^)1/Г- \r) r j'=i Since (ήα)η (α) = 0η(α), and (ήα)η (α) = 21п(а) the following result that contains an analogue of S(r;s) shows that these means form yet another scale of comparable means, between the geometric and arithmetic means. Theorem 1 (a) Let α be an η-tuple and r an integer, 1 < r < n, then 6Lrl(a)<(i5a)M(s), with equality if and only if either r = 1, or r = η or α is constant. (b) Let α be an η-tuple and r and s integers, 1 < r < s < n, then (Яа)1?](а)<(Яа)И(а), with equality if and only if α is constant. D (a) Immediate from (r;s). (b) (г) The first proof is simple and direct and is that of Hamy. If we assume that α is not constant it is sufficient to prove Σ!(ΓΚ)1/(9+1) < (™-9)Σ!(ΠΚ)1/9· w 9+1 .7=1 Я J = i

Means and Their Inequalities 365 Consider a typical term in the left-hand side sum. (a!a2...ag+1)1/(9+1) = {(a2a3 ... a^)1'*^ ... α^)1/* ... (αχα2 ... α,)1/^)1/(9+1) (а2аз · · - a<?+i)1/(? 4- (агаз · · · a<?+i)1/<? + ■ ■ ■ 4- (αια2 · · · a<?)1/<? b (QA^ 94-1 Applying this argument to each term of the sum on the left-hand side of (1) we find / \ 1 q that this sum is less than (д4-1)(д4-1)!( ) terms of the type τίΤΤ^ι)1^9· \Q 4-1/ 9 4-1 aa However there are only q\ ( ) different terms of this type, and by symmetry each w occurs equally often— that is (q -{- l)(n — q) times. This gives (1). (гг) An alternative proof was given in [Dunkel 1909/10]. By (r;s), assuming that a is not constant, ((**№(*»' = (^ή)ΣκΠ«,)1/9) > (^η)ΣκΠ%)1/(9+1 \ 9+1 ) ι By S(r;s) applied to a1/^4"1), we have that /i Я ч1/д / 1 9+1 \ l/(9+D Combining these two inequalities we get the required result. D The paper by Ku, Ku L· Zhang contains some interesting inequalities including the fact that ((ίο)Ρ(α))Γ is log-concave. 7.2 The Hayashi Means Interchanging products and sums in 1(2) lead to the means defined by Hayashi; [Hayashi]. ^(9) = ΚΠΚΣ^))(;). *^<η. Д") Theorem 2 If 1 < r < s then sii](a)<si?](a). with equality if and only if α is constant. D For a proof the reader is referred to the original paper. D

366 Chapter V Since Ίν(α) = ®n(a) and Xj^(a) = 2ln(&) this is yet another generalization of (GA). 7.3 The Biplanar Means Using the ideas of III 5.2.1 we can follow Gini and others to define what they called the biplanar combinatorial (p,q) power mean of order (c,d); here c, d are integers, 1 < c, d < n; see [Gini et al], [Castellano 1948,1950; Gatti 1956b,1957; Gini 1938; Zappa 1939]. Example (i) Clearly Ъ1^4 = 6^, and Q^'1^'1 = 0™. The following simple theorem is in [Gini L· Zappa]. Theorem 3 With the above notations r&l;d+m]1'd is a decreasing function of d, and *B^'c;1,d is a decreasing function of с D The first part is an immediate consequence of 2 Remark (iii). For the second part first note that «J^1·* = (Ul=d+i B^1'14-1)17^"^. So, by the first part <B^'C;1' is a geometric mean of an increasing sequence. Непсзе»^^»^1'0"1. Further <B^c;1'd = (B^^^-^i/Cc-dJ^.c-i-.ii.dji-i/Cc-d) showing that <B*'C;M is a geometric mean of the two means on the right-hand side. As we have just seen it is not less than the first of these two means and so, by the internality of the geometric mean, II 1.2 Theorem 5, it cannot exceed the second. D Remark (i) The proof of the second part of Theorem 3 shows that ш1,с;1,с-1 < Ж1|С;М < fBl,c-l;l,dj and it is easily deduced that if к > 0, с < d then b^+M.c-i < ^i.d < Remark (ii) A similar extension that is related to the Gini means, III 5.2.1, has been discussed in [Jecklin 1948b; Jecklin L· Eisenring]. 7.4 The Hypergeometric Mean Let α and b be η-tuples, u/ an (n - l)-tuple; let Ε be the set of y/ with the sum of the elements, Wn_i, less than 1. For each u/ £ Ε let w_ = (υ;', 1 —Wn_i), so that Wn = 1. Then for ρ G R the hypergeometric R-function is R(p,b,a)= / (^гУг^О VP{b,m)au/\

Means and Their Inequalities 367 Р(к,Ш.) = nn U(h TTT n^i_1' *s ^ne weight function, and fEP(b,w) &w! = 1· The hypergeometric R-function has a close connection with the theory of means. A (homogeneous) hypergeometric mean of α is constructed as follows; see [Carlson 1965,1966]. If c> 0, t(p,c,a,w) = ( (^(-№,α))ΐΛ\ if p^O, L limp_>0 £{P, c; α, υ;), if ρ = 0. The following result is in [Carlson L· Tobey). Theorem 4 (a) limc^0 £(p, c; a, w) = OJlJf] (α; υ;). (b) If α is not constant and ρ > 1, с > 0 then €(р,с;а,11;)<аЯЙ,1(а;ш). (2) (c) If α is not constant and ρ > 1 and с < с' then £(p,c;a,w) < €(p,c';a,w). (3) If ρ < 1 then inequalities (~2) and (~3) hold. Remark (i) Carlson has pointed out that the Whiteley means are special cases of the hypergeometric mean, and that the following generating relation is valid. Π(ΐ -ίαο— = f;c(c+1)--n{c+n"1)jz(-n,ca,a)t". Remark (ii) For further extensions see [Tobey].

VI OTHER TOPICS This chapter will cover a variety of topics that do not fit into the previous discussions. In particular there are two variable means, means defined for pairs of numbers and which do not not readily generalize to η-tuples. There is an elementary introduction to integral means and to matrix analogues of mean inequalities The topic of axiomatization of means is discussed but only briefly as the topic leads away from the interest of this book into the theory of functional equations and functional inequalities; [Aczel 1966]. 1 Integral Means and Their Inequalities 1.1 Generalities The extension of the concept of a mean to functions by the 1 fb use of integrals is a natural one. In most calculus texts the quantity / / b~a Ja is called the average, or arithmetic mean, of f on the interval [a,b], and this is justified by noting that the approximating Riemann sums are the arithmetic means of values of / at points distributed across [a, b]. In the usual notation, 1 n ^7^^/Ы(ж* -x%-i) = %п{а;ш), (1) г—ι where α^ = f(yi), m = (xi - Xi-i), 1 < г < n. Further developments of this idea use concepts outside the scope of this book as the natural setting is that of general measure spaces. However while most books on measure theory discuss certain inequalities, in particular integral analogues of (H), (J) and (M). they do not usually consider means. If we restrict ourselves to real-valued functions on an interval the approach in (1) can be used to extend (J). Then the definition of a quasi-arithmetic mean is obtained by analogy with the discrete case, and this leads to the basic mean inequalities. If the integrals used are of the Stieltjes type these integral results may imply the discrete results as special cases. This method does not yield the cases of equality, but these can be obtained under wide assumptions by the methods in [HLP p. 151}] see also [Julia], [Mitrinovic L· Vasic 1968a), and 1.3.1 below. 368

Means and Their Inequalities 369 Remark (i) Another approach that incudes both the integral and discrete inequalities as special cases is the use of time scales; see [Agarwal, Bohner L· Peterson]. More precisely: if μ : I = [a,b] ^ 1 is an increasing, left-continuous function1 with μ(Ι) = μ(6) — μ(α) > О2 then we say that / : [a, b] >—> Ж is μ-integrable on [a, b] if there is an I G IR such that for all e > 0 there is a δ : [α, b] ^ R+ such that for all J-fine partitions w = (ao, ai,.. . , an; yi,...,yn) such that η |]Γ/(1μ-ΐ| < б, where ^ /άμ = ]Γ /(^)(μ(α-0 - μία,.,)). (2) τσ ш г—ι Then I = Ι(/) is the μ-integral of f on I — [a, 6]. written I = Jf f άμ = Ja f άμ, or/ab/(^)Mdx).3 The exact integral defined depends on what is meant by a J-fine partition; in all cases α = ao < a\ < · · · < an = b\ if then [<ц-иаг] c]y\ -S(yi),yi-\-S(yi)[, 1 < г < η the integral is the Lebesgue-Stieltjes integral, if in addition уг Ε [ατ-1}α-ι\, 1 < г < η, it is the Perron-Stieltjes integral, and if in this case J is a constant, or just continuous, the integral is the Riemann-Stieltjes integral. Convention In the case μ(χ) = ж, а< х <b, when the measure is Lebesgue measure, the integral is written /a/, or faf(x)dx, and references to both μ and Stieltjes are omitted from all associated notations. If μ(Ι) = 1 then the measure is called a probability measure (on I). For details see [Bartle; Lee & Vyborny] where it is shown that the ideas extend easily to unbounded intervals. Which form of integration used does not matter, and in the case of μ-almost everywhere non-negative functions the more general Perron-Stieltjes integral is equivalent to the Lebesgue-Stieltjes integral. Since in considering inequalities we always require that / be μ-integrable and also that |/| be μ-integrable this will mean that our μ-integrals will be Lebesgue-Stieltjes integrals4, and we write / Ε £μ(α,6) IfpeM; we write /e££(a,b) if \f\p e £μ(α, b) and £j>,b) = £μ(α,6). We complete this definition by introducing the following quantities: the μ-essential These restrictions are convenient but arc not necessary; see [Bartle pp 391 399] In fact much more general measures can be considered; [Rudin pp.60 63], [Roselh L· Willem] 2 Note that this implies that 0<μ<οο. This corresponds to the assumption of non-negative weights in previous chapters and is only waived for 1.2 1 Theorem 2 below. 3 It will be convenient to allow an integral to be infinite when the integrand is non-negative almost everywhere; sec [Lieb & Loss ρ 15]. 4 However more general integrals have been used even though the hypotheses arc shown to imply Lebesgue integrability. sec [Ostaszcwski L· Sochacki]

370 Chapter VI upper bound off = inf{/3; μ{χ\ f(x) > β} = θ}, and the μ-essential lower bound of f — sup{/3; μ{χ] f(x) < β} = θ}; / is said to be μ-essentially bounded, /б£°°(а, 6), if both of these quantities are finite. In all these classes the functions are finite μ-almost everywhere. As usual if ρ e Ш* then p' is the conjugate index of p; Notations 4. Further we can, by considering the function 1//, allow ρ < 0, but we now must assume that / ψ 0 μ-almost everywhere 1 2 Basic Theorems 1.2.1 Jensen, Holder, Cauchy and Minkowski Inequalities The following inequality is an integral analogue of (J), and will be referred to as ( J)-/. There are many proofs in the standard literature; the one below is from [Lieb & Loss pp.38-39]; see also [HLP pp. 150-151; PPT pp45-47]. THEOREM 1 [Jensen's Inequality] If f £ £μ(α,6), and if Φ is convex and μ- integrable on some interval ]c,d[ containing the range of f then <mf.'^ml·"^ (JH If Φ is strictly convex then ( J)-J is strict unless f is μ-almost everywhere constant. 1 fb D That A = ——- / / dμ is in the domain of Φ is a simple consequence of β\* ) J а obvious properties of integrals, and the fact that the domain of Φ is as stated. Further since Φ is continuous the function Φ ο / is μ-measurable. By I 4.1 Corollary 5 Φ has a support at A; that is for some constant m, Ф{у) > Ф{А) + т(у -A),c<y<d (3) From (3) we see that (Фо/)-(ж) < |Ф(Л)|Ч-|ш||Л|Ч-|т||/(х)|, so (Φο/)~ Ε£μ(α, b) If right-hand side in (J)-/ is infinite the result is trivial so we can assume that Φο/Ε£μ(α,6). Now take у = f(x) in (3) to get i°*M>-*(w)Lfd>')+m{,ix,-inL>i>' (4) which on integrating gives (J)-/. If Φ is strictly convex then (3) is strict except when у — A. However if / is not constant μ-almost everywhere then (4) must be strict on a set of positive μ-measure, and so (J)-/ is strict. D

Means and Their Inequalities 371 Remark (i) The result also follows by applying (J) to the Riemann sums in (2), and taking limits, as in the discussion of (1) but this does not give the case of equality. An example of this procedure is given below in 1.3.1 Theorem 9. Remark (ii) As was suggested in the preliminary remarks Theorem 1 can be considered in more general settings; see [PPT pp.43-65), [Abramovich, Mond L· Pecaric 1997; Beesack &: Pecaric; Roselli &: Willem]. Remark (iii) If Φ" 0 then ( J)-/ is strict unless is -almost everywhere constant; see [HLP p.51). An integral analogue of the Jensen-Steffensen inequality, I 4.3 Theorem 20, can also be proved; [AIp.109;, MPFp.13]; [Boas 1970; Fink L· Jodeit 1990). A measure , not necessarily non-negative, on [ ] is said to be end-positive if ([ ]) 0 and for all < < ([ ]) > 0 and ([ ]) > 0 This is a generalization of I 4.3 Remark(i). THEOREM 2 [Jensen-Steffensen Inequality] If is an end-positive measure on [ ), e £μ( ) is monotonic, and if Φ is convex and -integrable on some interval containing the range of then ( J)-f holds. THEOREM 3 (a) [Holder's Inequality] If 1 : [ ] \—► R are -almost everywhere non-negative, with б£^( ), and Ε £Ρμ ( ), then e £μ( ), and if is not zero -almost everywhere there is equality if and only if for some Ε R we have that p = p -almost everywhere. (b) [Cauchy's Inequality] If : [ ] \—► R are -almost everywhere non-negative, with e£2( ), then Ε£μ( ), and if is not zero -almost everywhere there is equality if and only if for some e R we have that = -almost everywhere. (c) [Minkowski's Inequality] If 1, and if €£%( ) are -almost everywhere non-negative, then ( -h )e£^( ), and

372 Chapter VI if f is not zero μ-almost everywhere there is equality if and only if for some A e R we have that f = Xg μ-almost everywhere. D (a) Since the result is trivial if either / or g is μ-almost everywhere zero we may suppose that both are μ-almost everywhere positive. Then we can use proof (г) of III 2.1 Theorem 1 by considering, fg _ / ρ \l/P/ gP γ/Ρ' < Ρ' This implies that fg is /i-integrable, and integrating both sides gives (H)-/. The case of equality follows from that of (GA). (b) This is just the case ρ = ρ' = 2 of (a); the integral form of (C) is due to Bunyakovskii, [Bunyakovskif\. (c) The deduction of (M) from (H), III 2.4 Theorem 9 proof (г), can be adapted using (H)-J\ D Remark (iv) If ρ < 1 then we have (~H)-/, but the conditions of integrability have to be restated; [HLP pp. 140-141]. Remark (v) Using (H)-/ it is easy to show that if-oo<r<s<oo then / e £^(a, b) implies that / Ε £τμ(α, Ь); again care must be taken when we allow negative indices. As in the case of (M) an important use of (M)-/ is the proof of the triangle inequality; see III 2.4 (T). If f,g,he ££(a, b), ρ > 1 then, (Ja\f-9\P d/i)1/P < ( J \f - h? άμ)1/P + ( JЬ\Н - g\P άμ) У*. (T)- / l/p This is the essential property for showing that ||/||μ,ρ = ( Ja \f\pdμ) , ρ > 1, is a norm, on the class of functions ££(a, b).6 Example (i) The following is a simple inequality for the factorial function is application of (H)-/; [de la Vallee Poussin]. If ρ > 1, [0 < ρ < 1], 1 ί°° , , / ί°° \ !/ρ / Γ°° \ ι/ρ' (-)!=/ e-^^dx <,[>], Π e-xxdxj Π е"ж dx\ = 1. 5 More precisely: a norm on the equivalence classes of μ-almost everywhere equal functions of £^(a,b); [Lieb & Loss pp. 36-37; Rudin p.65]

Means and Their Inequalities 373 1.2.2 Mean Inequalities Suppose now that / : i" = [a, b] ι—► R, and that Μ is a strictly monotonic function defined on an interval containing the range of /, with Mo f e £μ(α, b) then the quasi-arithmetic M-mean, or just quasi-arithmetic mean, of f on [a, b] with weight μ is: OTM](/;M) = M-1(^7y^ Λίο/άμ); (5) and when convenient we will define 9Jt[a,a](/; μ) = /(a)· There is no loss in generality in assuming Μ to be strictly increasing; see IV 1.2 Remark (ii) Remark (i) The various usages associated with IV 1.1 Definition 1 will be used without further comments; in particular if μ is Lebesgue measure it is omitted from the notation. In addition if μ (da;) = τη(χ)άχ we will write 9Я[а,ь](/; μ) = ЯЯ[а,Ь](/;™). Remark (ii) These means can be defined on more general measure spaces, in particular for non-compact intervals, but that will not be pursued here; see most of the references and in particular [Losonczi 1977]. In addition the Bajraktarevic means, IV 7.1, have been studied in their integral form; [Gigante 1995b]. Remark (Hi) The quasi-arithmetic Л4-теап sometimes occurs with a different notation; see [HLP p.158} where / : R i-> R, f^ dμ = 1, and 9Jl(/; μ) = Μ-1(!Ζ0Μοίάμ). . Using the methods of IV 2 Theorem 5 the following result follows from Theorem 1. Theorem 4 Let Λ4 and λί be strictly monotonic functions on an interval that contains the range of f : I = [a, b] \—► R, Λί strictly increasing, [respectively decreasing], then for all f with Λ4 о /, TV о /e£M(a, b), ш[аМи^)<щаМи·^) (6) if and only if Λί is convex, [respectively concave], with respect to ΛΛ. Further if Λί is strictly convex, [respectively, strictly concave], with respect to ΛΛ then (6) is strict unless f is μ-almost everywhere constant. If Λί is decreasing, [respectively, increasing], and Λί is convex, [respectively, concave], with respect to Μ then (~6) holds. Remark (iv) If {ΛίοΛΛ-1)" > 0 then (6) is strict unless / is μ-almost everywhere constant; see 1.2.1 Remark (iii).

374 Chapter VI Taking M(x) = xr, r e R*, assuming that / > 0 μ-almost everywhere, / > 0 μ-almost everywhere if r < 0, and / e £τμ(α, Ь), (5) defines the r-tii power mean of/ on [a, b] with weight μ <4(/^) = (^/βΓ<1μ)1/Ρ; If r = 0 and logo/ ε £μ(α, Ь) then (5) defines the mean 9#L bi(/; μ). The case r = 0 is also the limit of the general case on letting r —► 0; see[#I/P 136-139]. The particular cases r = —1,0,1, 2 are as expected called the harmonic, geometric, arithmetic, and quadratic means of f on [a, b], with weight μ respectively, written: fl[a,b](/;/*)= J'I/fd » 6кь](/;м) = ехр(—— у logo/d/x); reference should be made to [ЖР pp. 136-137] for the exact interpretation of ^[a,b](/; μ)· Letting r -> ±oo we get 9Jt~? (/; μ) = μ-essential lower bound of / on [a, b]; ^ abl (^' ^) ~ μ-essential upper bound of / on [a, b]. If μ is Lebesgue measure we drop it from the notations, in accordance with the Convention in 1 above; this corresponds to the equal weight case for discrete means. Remark (v) These means have most of the properties of the analogous discrete means, as is easily seen. For instant the arithmetic mean has the appropriate properties, (Re), (Ho), (Ad), listed in II 1.1 Theorem 2. The properties (In) and (Mo) are given below in Theorem 6 (a), (b). Remark (vi) Another useful property is the analogue of III 2.3 Lemma 6(b); the function m(r) = rrif{r) = (ШТГ^/; μ))Γ, r G 1, is log-convex, strictly unless / is /i-almost everywhere constant; [HLP рр.145-Ц6]. The following simple example of invariance property, see II 5.2, V 2 Corollary 5, is often useful; see [AI p.9] Theorem 5 If f is strictly increasing on [a, b] then 2l[Xj2/](/; μ), α < χ < у < b, is strictly increasing in both χ and y. D Suppose that α < χ < у < ζ < Ь, and let I = [x,y], J = [y,z],K = [x,z], then simple calculations show that Я[х,*](/;м) = ^%,„](/;μ) + ^fivMf^)-

Means and Their Inequalities 375 Thus the integral arithmetic mean on the left-hand side is also an arithmetic mean of the two integral arithmetic means on the right-hand side. So the integral arithmetic mean on the left lies between the two integral arithmetic means on the right, and if / is non-negative and increasing we get from the internality of the discrete arithmetic mean, that, This implies the required result. D Remark (vii) A simple corollary of this result is that integral power means have the same property. Theorem 6 (a) [In] If -oo < r < oo then μ-essential lower bound of f < 9Л£bi(/; μ) < μ-essential lower bound of f. (7) (b) [Mo] If f < g μ-almost everywhere then Ч!ч^)^Ч!ы(^)· (c)[(r;s)] If — oo < r < s < oo then, Ч!ь](/;м)<Ч1ь](/^)' (T'>*H (d) [(GA)] й[а,ь](/;м) < в[а>ь](/;/х) <а[а>ь](/;/х), (GA)-/ In (a), (c) and (d) there is equality if and only iff is μ-almost everywhere constant. D (a) and (b) follow from simple properties of the integral. (c) (г) Taking particular functions for M. and Μ we get (r;s)-| for finite r and s from (6); this is completely analogous to the discrete case, see IV 2 Example (v). The other cases are included in (7). (гг) Of course there are many independent proofs of (r;s)-|, —oo < r < s < oo. Consider for instance the following; [Qi 1999b; Qi, Mei, Xia & Xu). Let I = [a, b] and define, for t φ Ο, φ{ί) = log Ы[^ь] (/; μ)). Then m _1оё(/аЬ/ММ)-1оё(М/)) = (log/ab/4M)-log(/abrdM) rb tJo V £ράμ J by interchanging differentiation and integration, [McShane pp.216-217].

376 Chapter VI By Theorem 6(b) the result follows if we can show that the integrand in the last f fs\ogof άμ integral, £(s) = ——^ , is increasing. Now, again interchanging differen- Saf8^ tiation and integration, which is non-negative by (C)-/. This shows that ζ is increasing and completes the proof. (d) Clearly (GA)-/ is a special case of (r; s)-/. D Remark (viii) Inequality (7) can be generalized in a manner due to Cauchy, see II 1.2 Lemma 4 (a). *-*fr)*m$ ***&)■ (8) These inequalities have been studied and generalized by several authors; [Kara- mata; Polya & Szego 1972 pp. 80,90], [Godunova 1972; Karamata; Kwon & Shon; Lukkassen; Lupa§ 1975,1978; Pecaric L· Savic; Sandor L· Toader; Winckler I860]. Remark (ix) The proof (гг) of (c) can be modified to give a proof of the discrete version of (r;s). An interesting inequality compares the geometric means of the arithmetic means of the derivatives of two functions with the arithmetic mean of the derivative of their geometric mean. Theorem 7 Let f,g,h : [a, b) —► Ш\, with f increasing, g,h continuously differ- entiable and g{a) = h{a),g(b) = h(b) then в2(%1ь](р,;/),ам](л,;/))<а[а1ь](в,(р,л);/). Remark (x) The case a = 0, b = 1, g(x) = x2p+1, h(x) = x2qJtl is due to Polya; the generalization is by Alzer. Further generalizations in which the geometric mean has been replaced by other means such as the Gini or Extended means, see 2.1.3 below, have also been given; [DIp.206]6, [Polya & Szego p.72], [Alzer 1990p; Pearce & Pecaric 1998]. This reference has a mistake; the derivative is taken inside the square root, instead of outside

Means and Their Inequalities 377 Remark (xi) A study of the iteration of the arithmetic mean has been made by Biickner; [Biickner]. Nanjundiah and Ky Fan type inequalities have also been studied; [Rassias pp.27-50), [Cizmesija L· Pecaric; Kwon 1996). An extremely important extension of the integral power means to analytic functions is due to Hardy. If / is analytic in {z\ \z\ = \регв\ < 1} and ifO<p<l,0< r < oo, then we define7 9Jt[rl(/;p) exp(^J\og+\f(peie)\d9\, if r = 0, 1/r ^f_ \f(pei9)\rdej , if0<r<oo, ^Ρ-π<θ<π |/(Р^)|, if Г = OO. These means clearly have all the properties of the power means above, having analogues of (Η)-/, (Μ)-/, and (r;s)-|. In addition the following important theorem can be proved. Theorem 8 Ж[г] (/; ρ) is an increasing function of p, 0 < ρ < 1. D The case r = oo follows from the maximum modulus theorem for analytic functions; for the other cases see [Rudin p.330). D These properties form the basis of an important area of mathematics known as Hardy spaces; see [EM4, pp.366-369). 1.3 Further Results 1.3.1 A General Result A procedure whereby integral inequalities are deduced from discrete inequalities can be stated in a fairly general way. Theorem 9 Let φ : R3 \-+ Ш be continuous, J an interval, JCR,F,G: JnE; jH" : J x J ι-» R, a,b η-tuples with elements in J, w_ an η-tuple, λ Ε R. If for all such a, b, w_, (η η η \ J2^F(ai),J2WiG№^WiH(ai'bi4 ^λ' (9) i=i i—i г—ι / then for all f,g: [a, b] i-> J with F о f,G о g,H(f,g) μ-integrable ф( f Fo/dft / Οο9άμ, ί Я(/,fl)dμ) >λ. (10) Using the notation of Notations 8; log+—max{log,0}.

378 Chapter VI D If η > 0, and и e R3 then there is an ε > 0 such that if \u — v\ < ε, then \Ф(и) -ψ(ν)\ <η· Now there is a δ : [α, b] ι-» R+ such that for all o-fine partitions vj of [a, b], £>ο/<Ιμ- / Fo/dJ<-^=, WGog^- f Gog /d/i < V5' ΣΗ(/,9)άμ- ί Η(/,9)άμ < л/5" Hence ν* /" Fo/άμ, ί Οο9άμ, ί Η(/,9)άμ" ч «/α «/α ν/α У (η η η \ ^ги»^(а»),^и;»С?(Ь»),^и;»Я(а»,Ь») Ι -τ/ г—ι г—ι i=i / >λ-77, which completes the proof. D Remark (i) This result can be given several obvious variants and extensions to functions φ of two variables, or of more than three variables. Example (i) Taking ip{x,y,z) = x1^y1^''-z, F(x) = xp, G{x) = xp\ H(x,y) = xy,X = 0, then (9) is (H) and (10) is (H)-/. Example (ii) Taking ψ(χ,ν,ζ) = xy, F(x) = xp, G{x) = —,f = g and λ = χ ((Μ + m)2)/4Mm, where 0 < m < f < Μ; then (~ 9) is the Kantorovic, or Schweitzer, inequality, III 4.1 (11), (12), and (~10) is the integral analogue i^X/V < (Af-f-m)2 AMm (И) Remark (ii) A completely different proof of (11), also a special case of a general method, can be found in the interesting paper [Rennie 1963]. Remark (iii) A general integral approach to the Kantorovic inequality can be found in [Clausing 1982]; see also [Diaz &; Metcalf 1964b; Lupas &; Hidaka]. 1.3.2 Beckenbach's Inequality; Beckenbach-Lorentz Inequality Analogues of these inequalities, III 2.5.5 Theorem 17 and III 2.5.7 Theorem 19 (c), have been proved directly; [Mond L· Pecaric 1995b; Zhuang 1993].

Means and Their Inequalities 379 Theorem 10 Suppose ρ > 1, f e ££(a, b), with f μ-almost everywhere non- negative, and д€С^ (a, b), with g μ-almost everywhere non-negative, а > 0, β > 0, 7 > 0, and deune h by h = {ад/βΥ^. Then (a + 7f>dM)1/p > (α + 7/>Φ01/ρ /? + 7/ab/pd/i β + ΊίΐΗ9άμ with equality if and only if f = h, μ almost everywhere. IfO<p< 1 then (~12) holds. D Obviously /г G £J(a, &), and the right-hand side of (12) is just: {<*№*''*(<*№/<*)>'+ Ί fig*'άμ)1/Ρ ( ,. , ή , ч-i/p' V » 1 = U-v /vpP + 7 / gP άμ\ . (13) (α/βγ/ρ(α(β/α)ρ'+ΊΙ°9Ρ'άμ) V Λ 7 Now: β + Ί f ϊ9άμ<β + Ί(Ι Ράμ)1'^! 9ρ'άμ)1/Ρ , by (H)-/, =α1/Ρ(/3α-1/ρ)+(7 Ζ* Ράμ)1,ν(η J g*'atf* f fb \1/P/ ,, /"Ь , \1/P' <{α + Ί J Γάμ) (οΓ*Ι*&> +7J д' άμ) , by (Η), which by (13) completes the proof of (11). The cases of equality follow from the cases of equality for (H)-/ and (H). D Theorem 11 Let p, f,g,h, a, /?, 7 be as in the previous theorem . Further assume that а--у /ab/pd/x>0 and α - Ί J* hp άμ > 0. Then (α-7/>άμ)1/ρ ^ (a-7f>dM)1/p> (14) β-ιίΙί9&μ β-ιΙαΗ9^μ with equality if and only if f = h, μ almost everywhere. D As in the proof of Theorem 10, h £ £ρμ{α, Ь), and we have that the right-hand side of (14) is just (α-Ρ'/ΡβΡ' - 7 f*gp' άμ) Ρ . By HI 2.5.7 Theorem 19(a), the Holder-Lorentz inequality, with η = 2, α^ = αλ/ρ, «2 = {Ί$ΙΡάμ)1,Ρ, h = α-νΡβ, b2 = {η fig*' άμ)1,ν' we have / fb U/P/ ,, , fb , xl/p' (α - 7у /P<1μ) [ο? Ι*β* - 7у 3Р άμ) <P-l(J Γάμ)1/Ρ(Ι 9ρ'άμ)1/Ρ <β-Ί Ι Ϊ9άμ, by (Η)-/. J α

380 Chapter VI As in Theorem 10 this completes the proof. D 1.3.3 Converse Inequalities Following the general remarks above converse inequalities can be obtained from the discrete analogues but considerable work has been done on direct approaches; [Alzer 199lh; Barnes 1969; Choi; D'Apuzzo &; Sbordone; He L; Kwon 1995; Mond & Pecaric 1994; Nehari; Pecaric Sz Pearce; Yang & Teng; Zhuang 1991). The following converse of (J)-/ goes back to Knopp and should be compared to I 4.4 Theorem 28; [Popoviciu pp.33-35), [Knopp 1935; Pecaric L· Beesack 1987b). Theorem 12 Let Φ : [га, Μ] ι-» Ш be convex and strictly monotonic, and let д : [0,1] ь+ [га, М) then ί Фод-Ф([ д)<(1- *0)Ф(т) + ί0Φ(Μ) - Ф(Г^т + t0M), Jo Jo where 1 — ί0πι 4- toM is the mean-value point for Φ on [ra, Μ). A particularly simple example of a converse Holder inequality is given in the next theorem; [Zhuang 1993). Theorem 13 Iff, д : [α, Ь] ^ R with 0 < mx < f < Mb 0 < ra2 < д < M2, with fp and gp μ-integrable, ρ > 1, then (| Γάμ)1/Ρ(Ι /dM)1/P <K J /fld/i, (15) fmyp + Mj'/p' Ml/p + mi/p'\ where К = max < — , '—— >. \ raiM2 M1m2 J D Κ ί fдάμ> ί (-fp + ^gp') d/i, by II 4.2 Theorem 7, J α J α Ρ Ρ = - [ Ράμ + — ί gv' άμ > the left-hand side of (15), by (GA). Ρ J α Ρ J а D A converse of Beckenbach's inequality, 1.3.2 Theorem 10, has also been given by Zhuang. 1.3.4 Ryff 's Inequality In a very interesting paper Ryff, [Ryff], has discussed an integral analogue of Muirhead's theorem, V 6 Theorem 1, and of V 6 Corollary 2(a).

Means and Their Inequalities 381 For simplicity we consider real-valued functions on [0,1], and first we define an integral analogue to the order -< defined in I 3.3; see [A I pp. 162-168; DI pp. 9,198; HLP pp.276-279; MO p. 15; PPT pp.324-325]. If / and g are two decreasing functions we say the / -<; g when //</%, 0<x<l, and [ f= f g. (16) JO JO JO JO In general / -<; g if (16) holds between the decreasing rearrangements of / and g, where the decreasing rearrangement of a function /, written /*, is defined by requiring: (i) /* is decreasing and right continuous, (ii) the sets {x\ f{x) > y} and {x; f*(x) > y} have the same measure for all y. Various of the results for the order defined in 1.3.3 have analogues in this situation; in particular I 4.1 Theorem 10(a), and the following result of Ryff that is an analogue of Muirhead's theorem. THEOREM 14 [Ryff's Inequality] Let f,g be bounded and measurable on [0,1]. If f -< g , and if и is a positive function with up integrable for all ρ еШ, then J bg ( J u(y)№ dy) dx < J log ( J u{yY^ dy) dy. (17) Conversely if (17) holds for all such и then f -< д. Remark (i) The argument that (17) is the correct analogue of V 6(3) is given in detail in the paper of Ryff. If the order of integration in (17) is reversed the integrals may fail to exist. Remark (ii) The question of the cases of equality in (17) remains open. Ryff conjectures that there is equality if and only if either и is constant almost everywhere, or/* =g\ 1.3.5 Best Possible inequalities An inequality between two quantities P, Q of the form Ρ < KQ is usually understood to be best possible when K, a constant depending on the parameters in Ρ and Q, has been evaluated, and for no smaller constant does the inequality hold in general. In interesting papers Fink and Jodheit have suggested an alternative way of considering an inequality to be best possible; [Fink 1994; Fink L· Jodheit 1984,1990). This idea is best explained by the inequality of Karamata, I 4.1 Theorem 10(a): ΣΓ=ι f(bi) ^ ΣΓ=ι /(αΌ f°r aU functions f convex on I if and only ifb -< a. Fink pointed out that the following is also true: Σ™=1 f(pi) < Σ™=1 ί{ο>%) holds for all a,b with b^ a if and only if f is convex; [Fink 1994),

382 Chapter VI In other words Karamata's theorem not only characterizes the sequences for which the inequality holds for all convex functions, but also it characterizes the class of convex functions. We illustrate this idea further with two results, an integral analogue of Cebisev's inequality, see II 5.3, and (J)-/. We say that two functions defined on the interval [a, b] are similarly ordered when (f(x)—f(y)) {g(x)—g(y)) > 0 f°r all ж, уе[а, b\] this is analogous to I 3.3 Definition 15. In the next two theorems μ is a regular Borel measure8, and the notation in Theorem 16 is that of 1.2.1 Theorem 1. THEOREM 15 [Cebisev's Inequality] if f,g are piecewise continuous and non- negative μ-almost everywhere on [a, b], then the inequality pb nb pb pb / /pd/x / d/x > / /d/x / дάμ Ja J a J a J a holds for all similarly ordered pairs of functions f and д if and only if μ > 0. Further this inequality holds for all μ > 0. if and only if the functions f and д ar