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Handbook of Means and Their Inequalities

P. S. Bullen
Publisher: 
Kluwer Academic Publishers
Publication Date: 
2003
Number of Pages: 
537
Format: 
Hardcover
Series: 
Mathematics and Its Applications 560
Price: 
139.00
ISBN: 
978-1-4020-1522-9
Category: 
Handbook
[Reviewed by
Allen Stenger
, on
03/12/2011
]

This is an encyclopedia of inequalities related to means, very broadly defined. Despite the ordering of topics in the title, the book is nearly all about inequalities, and few other properties of means are considered. The book covers not only such old standbys as the arithmetic and geometric means, but power means, symmetric means (as in Muirhead’s theorem), and many kinds of norms such as those used in the Cauchy-Schwarz, Hölder, and Minkowski inequalities. It deals almost exclusively with discrete means (that is, finite sums), and the integral analogs of these and a few other integral-related results are confined to a separate 16-page section. The book is a thorough reworking of Bullen & Mitrinovic & Vasic’s Means and Their Inequalities (Kluwer, 1988), and is similar in style but more comprehensive than Mitrinovic’s classic Analytic Inequalities (Springer, 1970).

The author says “This book is aimed at a wide audience” (p. xxvi), and it opens with a 60-page introduction to the subject of inequalities that deals with some of the more elementary results. Proofs are given for all central results in the book, even to the point of overkill in some cases (there are 74 proofs of the Arithmetic Mean–Geometric Mean inequality, which makes the 7 proofs of Hölder’s inequality and 5 proofs of Minkowski’s inequality seem restrained). Less-central results are quoted and given references in the literature. Despite the introduction, it’s probably not a good book for beginners, because it simply has too much information and you don’t know where to start; reading it cover to cover is not the right answer. Better books for beginners are Beckenbach & Bellman’s An Introduction to Inequalities (for absolute beginners) or Steele’s The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (at the upper undergraduate level).

The author has done a good job of taming the notation, and the book is clearly written. It’s easy to find things in, being extensively cross-referenced and having both a detailed index and a surprisingly useful and detailed table of contents. There are a few peculiarities in the organization. For example, frequently-referenced books are given one- to three-character abbreviations, but the list of these is not in the bibliography where you would expect it but in the front matter.

Bottom line: Harald Bohr is reported to have said, “All analysts spend half their time hunting through the literature for inequalities they want to use, but cannot prove.” If you are in this category, or are an aficionado of inequalities, you will want to own a copy of this book, and it is a valuable reference for university libraries.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

  • PREFACE TO “MEANS AND THEIR INEQUALITIES”
  • PREFACE TO THE HANDBOOK
  • BASIC REFERENCES
  • NOTATIONS
    1. Referencing
    2. Bibliographic References
    3. Symbols for Some Important Inequalities
    4. Numbers, Sets and Set Functions
    5. Intervals
    6. n-tuples
    7. Matrices
    8. Functions
    9. Various
  • A LIST OF SYMBOLS
  • AN INTRODUCTORY SURVEY
  • CHAPTER I INTRODUCTION
    1. Properties of Polynomials
      • 1.1 Some Basic Results
      • 1.2 Some Special Polynomials
    2. Elementary Inequalities
      • 2.1 Bernoulli’s Inequality
      • 2.2 Inequalities Involving Some Elementary Functions
    3. Properties of Sequences
      • 3.1 Convexity and Bounded Variation of Sequences
      • 3.2 Log-convexity of Sequences
      • 3.3 An Order Relation for Sequences
    4. Convex Functions
      • 4.1 Convex Functions of Single Variable
      • 4.2 Jensen’s Inequality
      • 4.3 The Jensen-Steffensen Inequality
      • 4.4 Reverse and Converse Jensen Inequalities
      • 4.5 Other Forms of Convexity
        • 4.5.1 Mid-point Convexity
        • 4.5.2 Log-convexity
        • 4.5.3 A Function Convex with respect to Another Function
      • 4.6 Convex Functions of Several Variables
      • 4.7 Higher Order Convexity
      • 4.8 Schur Convexity
      • 4.9 Matrix Convexity
  • CHAPTER II THE ARITHMETIC, GEOMETRIC AND HARMONIC MEANS
    1. Definitions and Simple Properties
      • 1.1 The Arithmetic Mean
      • 1.2 The Geometric and Harmonic Means
      • 1.3 Some Interpretations and Applications
      • 1.3.1 A Geometric Interpretation
      • 1.3.2 Arithmetic and Harmonic Means in Terms of Errors
      • 1.3.3 Averages in Statistics and Probability
      • 1.3.4 Averages in Statics and Dynamics
      • 1.3.5 Extracting Square Roots
      • 1.3.6 Cesàro Means
      • 1.3.7 Means in Fair Voting
      • 1.3.8 Method of Least Squares
      • 1.3.9 The Zeros of a Complex Polynomial
    2. The Geometric Mean-Arithmetic Mean Inequality
      • 2.1 Statement of the Theorem
      • 2.2 Some Preliminary Results
        • 2.2.1 (GA) with n = 2 and Equal Weights
        • 2.2.2 (GA) with n = 2, the General Case
        • 2.2.3 The Equal Weight Case Suffices
        • 2.2.4 Cauchy’s Backward Induction
      • 2.3 Some Geometrical Interpretations
      • 2.4 Proofs of the Geometric Mean-Arithmetic Mean Inequality
        • 2.4.1 Proofs Published Prior to 1901. Proofs (i)-(vii)
        • 2.4.2 Proofs Published Between 1901 and 1934. Proofs (viii)-(xvi)
        • 2.4.3 Proofs Published Between 1935 and 1965. Proofs (xvii)-(xxxi)
        • 2.4.4 Proofs Published Between 1966 and 1970. Proofs (xxxii)-(xxxvii)
        • 2.4.5 Proofs Published Between 1971 and 1988. Proofs (xxxviii)-(lxii)
        • 2.4.6 Proofs Published After 1988. Proofs (lxiii)-(lxxiv)
        • 2.4.7 Proofs Published In Journals Not Available to the Author
      • 2.5 Applications of the Geometric Mean-Arithmetic Mean Inequality
        • 2.5.1 Calculus Problems
        • 2.5.2 Population Mathematics
        • 2.5.3 Proving other Inequalities
        • 2.5.4 Probabilistic Applications
    3. Refinements of the Geometric Mean-Arithmetic Mean Inequality
      • 3.1 The Inequalities of Rado and Popoviciu
      • 3.2 Extensions of the Inequalities of Rado and Popoviciu
        • 3.2.1 Means with Different Weights
        • 3.2.2 Index Set Extensions
      • 3.3 A Limit Theorem of Everitt
      • 3.4 Nanjundiah’s Inequalities
      • 3.5 Kober-Diananda Inequalities
      • 3.6 Redheffer’s Recurrent Inequalities
      • 3.7 The Geometric Mean-Arithmetic Mean Inequality with General Weights
      • 3.8 Other Refinements of Geometric Mean-Arithmetic Mean Inequality
    4. Converse Inequalities
      • 4.1 Bounds for the Differences of the Means
      • 4.2 Bounds for the Ratios of the Means
    5. Some Miscellaneous Results
      • 5.1 An Inductive Definition of the Arithmetic Mean
      • 5.2 An Invariance Property
      • 5.3 Cebisev’s Inequality
      • 5.4 A Result of Diananda
      • 5.5 Intercalated Means
      • 5.6 Zeros of a Polynomial and its Derivative
      • 5.7 Nanson’s Inequality
      • 5.8 The Pseudo Arithmetic Means and Pseudo Geometric Means
      • 5.9 An Inequality Due to Mercer
  • CHAPTER III THE POWER MEANS
    1. Definitions and Simple Properties
    2. Sums of Powers
      • 2.1 Hölder’s Inequality
      • 2.2 Cauchy’s Inequality
      • 2.3 Power sums
      • 2.4 Minkowski’s Inequality
      • 2.5 Refinements of the Hölder, Cauchy and Minkowski Inequalities
        • 2.5.1 A Rado type Refinement
        • 2.5.2 Index Set Extensions
        • 2.5.3 An Extension of Kober-Diananda Type
        • 2.5.4 A Continuum of Extensions
        • 2.5.5 Beckenbach’s Inequalities
        • 2.5.6 Ostrowski’s Inequality
        • 2.5.7 Aczél-Lorentz Inequalities
        • 2.5.8 Various Results
    3. Inequalities Between the Power Means
      • 3.1 The Power Mean Inequality
        • 3.1.1 The Basic Result
        • 3.1.2 Hölder’s Inequality Again
        • 3.1.3 Minkowski’s Inequality Again
        • 3.1.4 Cebisev’s Inequality
      • 3.2 Refinements of the Power Mean Inequality
        • 3.2.1 The Power Mean Inequality with General Weights
        • 3.2.2 Different Weight Extension
        • 3.2.3 Extensions or the Rado-Popoviciu Type
        • 3.2.4 Index Set Extensions
        • 3.2.5 The Limit Theorem of Everitt
        • 3.2.6 Nanjundiah’s Inequalities
    4. Converse Inequalities
      • 4.1 Ratios of Power Means
      • 4.2 Differences of Power Means
      • 4.3 Converses of the Cauchy, Hölder and Minkowski Inequalities
    5. Other Means Defined Using Powers
      • 5.1 Counter-Harmonic Means
      • 5.2 Generalizations of the Counter-Harmonic Means
        • 5.2.1 Gini Means
        • 5.2.2 Bonferroni Means
        • 5.2.3 Generalized Power Means
      • 5.3 Mixed Means
    6. Some Other Results
      • 6.1 Means on the Move
      • 6.2 Hlawka-type inequalities
      • 6.3 p-Mean Convexity
      • 6.4 Various Results
  • CHAPTER IV QUASI-ARITHMETIC MEANS
    1. Definitions and Basic Properties
      • 1.1 The Definition and Examples
      • 1.2 Equivalent Quasi-arithmetic Means
    2. Comparable Means and Functions
    3. Results of Rado-Popoviciu Type
      • 3.1 Some General Inequalities
      • 3.2 Some Applications of the General Inequalities
    4. Further Inequalities
      • 4.1. Cakalov’s Inequality
      • 4.2 A Theorem of Godunova
      • 4.3 A Problem of Oppenheim
      • 4.4 Ky Fan’s Inequality
      • 4.5 Means on the Move
    5. Generalizations of the Holder and Minkowski Inequalities
    6. Converse Inequalities
    7. Generalizations of the Quasi-arithmetic Means
      • 7.1 A Mean of Bajraktarevic
      • 7.2 Further Results
      • 7.2.1 Deviation Means
      • 7.2.2 Essential Inequalities
      • 7.2.3 Conjugate Means
      • 7.2.4 Sensitivity of Means
  • CHAPTER V SYMMETRIC POLYNOMIAL MEANS
    1. Elementary Symmetric Polynomials and Their Means
    2. The Fundamental Inequalities
    3. Extensions of S(r;s) of Rado-Popoviciu Type
    4. The Inequalities of Marcus & Lopes
    5. Complete Symmetric Polynomial Means; Whiteley Means
      • 5.1 The Complete Symmetric Polynomial Means
      • 5.2 The Whiteley Means
      • 5.3 Some Forms of Whiteley
      • 5.4 Elementary Symmetric Polynomial Means as Mixed Means
    6. The Muirhead Means
    7. Further Generalizations
      • 7.1 The Hamy Means
      • 7.2 The Hayashi Means
      • 7.3 The Biplanar Means
      • 7.4 The Hypergeometric Mean
  • CHAPTER VI OTHER TOPICS
    1. Integral Means and Their Inequalities
      • 1.1 Generalities
      • 1.2. Basic Theorems
        • 1.2.1 Jensen, Hölder, Cauchy and Minkowski Inequalities
        • 1.2.2 Mean Inequalities
      • 1.3 Further Results
        • 1.3.1 A General Result
        • 1.3.2 Beckenbach’s Inequality; Beckenbach-Lorentz Inequality
        • 1.3.3 Converse Inequalities
        • 1.3.4 Ryff’s Inequality
        • 1.3.5 Best Possible Inequalities
        • 1.3.6 Other Results
    2. Two Variable Means
      • 2.1 The Generalized Logarithmic and Extended Means
        • 2.1.1 The Generalized Logarithmic Means
        • 2.1.2 Weighted Logarithmic Means of n-tuples
        • 2.1.3 The Extended Means
        • 2.1.4 Heronian, Centroidal and Neo-Pythagorean Means
        • 2.1.5 Some Means of Haruki and Rassias
      • Mean Value Means
        • 2.2.1 Lagrangian Means
        • 2.2.2 Cauchy Means
      • 2.3 Means and Graphs
        • 2.3.1 Alignment Chart Means
        • 2.3.2 Functionally Related Means
      • 2.4 Taylor Remainder Means
      • 2.5 Decomposition of Means
    3. Compounding of Means
      • 3.1 Compound means
      • 3.2 The Arithmetico-geometric Mean and Variants
        • 3.2.1 The Gaussian Iteration
        • 3.2.2 Other Iterations
    4. Some General Approaches to Means
      • 4.1 Level Surface Means
      • 4.2 Corresponding Means
      • 4.3 A Mean of Galvani
      • 4.4 Admissible Means of Bauer
      • 4.5 Segre Functions
      • 4.6 Entropic Means
    5. Mean Inequalities for Matrices
    6. Axiomatization of Means
  • BIBLIOGRAPHY
    • Books
    • Papers
  • NAME INDEX
  • INDEX

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