Handbook of Means and Their Inequalities

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.