Synopses & Reviews
This is a revision of an earlier
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 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.
Audience: This book is written in a language that not only the expert on the subject will understand and appreciate, but graduate students worldwide as well. 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.
Review
"This book is indeed a handbook: it is hard to find a subfield of inequalities related to means that is not described in detail, or at least referred to. It is written in a language that can be understood not only by experts but also by graduate students and others who are interested in the applications: the only requirement on readers is a knowledge of calculus and elementary real analysis."--MATHEMATICAL REVIEWS
Synopsis
This is a revision of an earlier 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 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. Audience: This book is written in a language that not only the expert on the subject will understand and appreciate, but graduate students worldwide as well. 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.
Synopsis
This book updates its earlier version and enlarges the scope considerably to give a full and in-depth treatment the topic of means and their inequalities. Among its features are a variety of proofs for many of the basic results, and an extensive bibliography.
Table of Contents
- 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. - I: Introduction. 1. Properties of Polynomials. 2. Elementary Inequalities. 3. Properties of Sequences. 4. Convex Functions. - II: The Arithmetic, Geometric and Harmonic Means. 1. Definitions and Simple Properties. 2. The Geometric Mean-Arithmetic Mean Inequality. 3. Refinements of the Geometric Mean-Arithmetic Mean Inequality. 4. Converse Inequalities. 5. Some Miscellaneous Results. - III: The Power Means. 1. Definitions and Simple Properties. 2. Sums of Powers. 3. Inequalities between the Power Means. 4. Converse Inequalities. 5. Other Means Defined Using Powers. 6. Some Other Results. - IV: Quasi-Arithmetic Means. 1. Definitions and Basic Properties. 2. Comparable Means and Functions. 3. Results of Rado Popoviciu Type. 4. Further Inequalities. 5. Generalizations of the Hölder and Minkowski Inequalities. 6. Converse Inequalities. 7. Generalizations of the Quasi-arithmetic Means. - 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 and Lopes. 5. Complete Symmetric Polynomial Means: Whiteley Means. 6. The Muirhead Means. 7. Further Generalizations. - VI: Other Topics. 1. Integral Means and Their Inequalities. 2. Two Variable Means. 3. Compounding of Means. 4. Some General Approaches to Means. 5. Mean Inequalities for Matrices. 6. Axiomatization of Means. Bibliography. Name Index. Index.