Handbook of Means and Their Inequalities / Edition 2

Handbook of Means and Their Inequalities / Edition 2

by P.S. Bullen

ISBN-10: 1402015224

ISBN-13: 9781402015229

Pub. Date: 12/31/1987

Publisher: Springer Netherlands

This is a revision of an earlier work Means and Their Inequalities by the present author and Professors Mitrinovic and Vasic. Not only does this book bring the earlier version up to date but it enlarges the scope considerably to give a full and in-depth treatment of all aspects of the field. While the mention of means occurs in many books this is the only full


This is a revision of an earlier work Means and Their Inequalities by the present author and Professors Mitrinovic and Vasic. Not only does this book bring the earlier version up to date but it enlarges the scope considerably to give a full and in-depth treatment of all aspects of the field. While the mention of means occurs in many books this is the only full treatment of the subject. Outstanding features of the book are the variety of proofs given for many of the basic results, over seventy for the inequality between the arithmetic and geometric means for instance, an exhaustive bibliography and a list of mathematicians who have contributed to this field from the time of Euclid to the present day. The results discussed and the proofs provided are written in a language that not only the expert in the subject will understand but also graduate students worldwide will understand and appreciate. Any person with an interest in means and their inequalities should find this book within their comprehension although to fully appreciate all the topics covered a knowledge of calculus and of elementary real analysis is required.

Product Details

Springer Netherlands
Publication date:
Mathematics and Its Applications (closed) Series, #560
Edition description:
2nd ed. 2003
Product dimensions:
8.27(w) x 11.69(h) x 0.05(d)

Related Subjects

Table of Contents

Preface to "Means and Their Inequalities"xiii
Preface to the Handbookxv
Basic Referencesxvii
2.Bibliographic Referencesxix
3.Symbols for Some Important Inequalitiesxix
4.Numbers, Sets and Set Functionsxx
A List of Symbolsxxiv
An Introductory Surveyxxvi
Chapter IIntroduction1
1.Properties of Polynomials1
1.1Some Basic Results1
1.2Some Special Polynomials3
2.Elementary Inequalities4
2.1Bernoulli's Inequality4
2.2Inequalities Involving Some Elementary Functions6
3.Properties of Sequences11
3.1Convexity and Bounded Variation of Sequences11
3.2Log-convexity of Sequences16
3.3An Order Relation for Sequences21
4.Convex Functions25
4.1Convex Functions of Single Variable25
4.2Jensen's Inequality30
4.3The Jensen-Steffensen Inequality37
4.4Reverse and Converse Jensen Inequalities43
4.5Other Forms of Convexity48
4.6Convex Functions of Several Variables50
4.7Higher Order Convexity54
4.8Schur Convexity57
4.9Matrix Convexity58
Chapter IIThe Arithmetic, Geometric and Harmonic Means60
1.Definitions and Simple Properties60
1.1The Arithmetic Mean60
1.2The Geometric and Harmonic Means64
1.3Some Interpretations and Applications66
2.The Geometric Mean-Arithmetic Mean Inequality71
2.1Statement of the Theorem71
2.2Some Preliminary Results72
2.3Some Geometrical Interpretations82
2.4Proofs of the Geometric Mean-Arithmetic Mean Inequality84
2.5Applications of the Geometric Mean-Arithmetic Mean Inequality119
3.Refinements of the Geometric Mean-Arithmetic Mean Inequality125
3.1The Inequalities of Rado and Popoviciu125
3.2Extensions of the Inequalities of Rado and Popoviciu129
3.3A Limit Theorem of Everitt133
3.4Nanjundiah's Inequalities136
3.5Kober-Diananda Inequalities141
3.6Redheffer's Recurrent Inequalities145
3.7The Geometric Mean-Arithmetic Mean Inequality with General Weights148
3.8Other Refinements of Geometric Mean-Arithmetic Mean Inequality149
4.Converse Inequalities154
4.1Bounds for the Differences of the Means154
4.2Bounds for the Ratios of the Means157
5.Some Miscellaneous Results160
5.1An Inductive Definition of the Arithmetic Mean160
5.2An Invariance Property160
5.3Cebisev's Inequality161
5.4A Result of Diananda165
5.5Intercalated Means166
5.6Zeros of a Polynomial and Its Derivative170
5.7Nanson's Inequality170
5.8The Pseudo Arithmetic Means and Pseudo Geometric Means171
5.9An Inequality Due to Mercer174
Chapter IIIThe Power Means175
1.Definitions and Simple Properties175
2.Sums of Powers178
2.1Holder's Inequality178
2.2Cauchy's Inequality183
2.3Power sums185
2.4Minkowski's Inequality189
2.5Refinements of the Holder, Cauchy and Minkowski Inequalities192
3.Inequalities Between the Power Means202
3.1The Power Mean Inequality202
3.2Refinements of the Power Mean Inequality216
4.Converse Inequalities229
4.1Ratios of Power Means230
4.2Differences of Power Means238
4.3Converses of the Cauchy, Holder and Minkowski Inequalities240
5.Other Means Defined Using Powers245
5.1Counter-Harmonic Means245
5.2Generalizations of the Counter-Harmonic Means248
5.3Mixed Means253
6.Some Other Results256
6.1Means on the Move256
6.2Hlawka-type inequalities258
6.3p-Mean Convexity260
6.4Various Results260
Chapter IVQuasi-Arithmetic Means266
1.Definitions and Basic Properties266
1.1The Definition and Examples266
1.2Equivalent Quasi-arithmetic Means271
2.Comparable Means and Functions273
3.Results of Rado-Popoviciu Type280
3.1Some General Inequalities280
3.2Some Applications of the General Inequalities282
4Further Inequalities285
4.1Cakalov's Inequality286
4.2A Theorem of Godunova288
4.3A Problem of Oppenheim290
4.4Ky Fan's Inequality294
4.5Means on the Move298
5.Generalizations of the Holder and Minkowski Inequalities299
6.Converse Inequalities307
7.Generalizations of the Quasi-arithmetic Means310
7.1A Mean of Bajraktarevic310
7.2Further Results316
Chapter VSymmetric Polynomial Means321
1.Elementary Symmetric Polynomials and Their Means321
2.The Fundamental Inequalities324
3.Extensions of S(r;s) of Rado-Popoviciu Type334
4.The Inequalities of Marcus & Lopes338
5.Complete Symmetric Polynomial Means; Whiteley Means341
5.1The Complete Symmetric Polynomial Means341
5.2The Whiteley Means343
5.3Some Forms of Whiteley349
5.4Elementary Symmetric Polynomial Means as Mixed Means356
6.The Muirhead Means357
7.Further Generalizations364
7.1The Hamy Means364
7.2The Hayashi Means365
7.3The Biplanar Means366
7.4The Hypergeometric Mean366
Chapter VIOther Topics368
1.Integral Means and Their Inequalities368
1.2Basic Theorems370
1.3Further Results377
2.Two Variable Means384
2.1The Generalized Logarithmic and Extended Means385
2.2Mean Value Means403
2.3Means and Graphs406
2.4Taylor Remainder Means409
2.5Decomposition of Means412
3.Compounding of Means413
3.1Compound means413
3.2The Arithmetico-geometric Mean and Variants417
4.Some General Approaches to Means420
4.1Level Surface Means420
4.2Corresponding Means422
4.3A Mean of Galvani423
4.4Admissible Means of Bauer423
4.5Segre Functions425
4.6Entropic Means427
5.Mean Inequalities for Matrices429
6.Axiomatization of Means435
Name Index511

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