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G. H. Hardy, J. E. Littlewood, and G. Pólya
Cambridge University Press
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Cambridge Mathematical Library
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This book (which we will abbreviate as HLP) is an aged and classic but still useful monograph on inequalities. We will survey it and compare it to its primary modern rival, Steele’s 2004 The Cauchy-Schwarz Master Class.

HLP was written in 1934, and the second edition in 1952 corrected a few things in the body and added three appendices with newer proofs of the three of the results in the body. Thus, we are dealing with an essentially 80-year-old book. It was the first systematic study of inequalities: before this, inequalities was not really “a subject”; individual inequalities appeared in fragmentary form buried in papers on other subjects. HLP made a deliberate effort to unify these fragments, and to present each inequality in its most general form. The exposition is very clear, and although it has the form of a monograph, it would have worked well as a textbook as well. The book doesn’t have any exercises or examples per se, but each chapter does end with a large collection of theorems stated without proof, some of which are specializations of theorems in the body. The prerequisites are low: a modest amount of calculus and ideally some Lebesgue integration. The proofs depend primarily on algebraic calculations, often intricate, but not conceptually difficult.

The authors considered that Chapters 2–6 represented the main exposition, with the remaining chapters being “essays” (special topics) that were not as systematic or complete. In fact Chapter 2 contains most of today’s mainstream inequalities: arithmetic mean-geometric mean, Cauchy-Schwarz, Hölder, Minkowski, Chebyshev, and Muirhead. There are presented only in their finite-sum forms in this chapter, because the authors emphasize (correctly) the importance of identifying the cases of equality in inequalities. The naive approach would be to take limits of these finite forms to get the analogous infinite-sum and integral inequalities, but we usually weaken strict inequalities to “less than or equal” by taking the limit, and the authors did not want to do this. Instead the infinite sums and the integrals are handled in their own chapters; in many cases the same proofs go through but in others special reasoning is needed to get the cases of equality.

Despite the care the authors obviously took in the exposition, it takes a lot of work to study or reference this book. The most serious difficulty is that there is no index or symbol index, so it’s hard to find things. The Table of Contents is very detailed, but many of the section headings are vague in the classic British manner, for example “2.23 A note on definite forms”. Coupled with this, the notation for means was not (and is still not today) standardized, so it’s hard to dip into the book and get a result; you don’t know what the symbols mean unless you’ve read the book up to that point. It’s also hard to scan the book looking for a relevant result unless you already know the notations.

The generality of the results is not as much of a barrier as you might think; everything is stated with arbitrary weights, and often you just want two terms with equal weights, but it’s usually not hard to figure out the specialization, and the general form is only slightly harder to prove. One section where the generality doesn’t work well is Chapter 3, on convexity; it backs into convexity by considering what desirable general properties inequalities might have, and then after several very general pages suddenly discovers that the already-familiar notion of convexity is what we want.

Steele’s The Cauchy-Schwarz Master Class covers roughly the same topics as HLP, although it is intended as a sampler and is not as comprehensive. It does cover a number of newer topics not in HLP, and convexity plays a much more central role here (it is one of Steele’s “three pillars” of the theory of inequalities, along with positivity and monotonicity). Steele also has many specific inequalities to be attacked through use of general equalities. In general the book takes a problem-solving approach rather than giving a systematic exposition (it is in the MAA Problem Books series), and all problems are solved in the body or the back of the book. The organization, including the excellent index, makes it much easier to use than HLP. Despite its scattershot nature, it’s still packed with useful information, and today it’s my go-to book for any kind of inequality problem.

Bottom line: HLP is still a valuable book, and has many results that are hard to find elsewhere. But it’s not the place to start learning inequalities: use Steele or one of the simpler books such as Kazarinoff’s Analytic Inequalities or Beckenbach & Bellman’s An Introduction to Inequalities.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

1. Introduction

2. Elementary mean values

3. Mean values with an arbitrary function and the theory of convex functions

4. Various applications of the calculus

5. Infinite series

6. Integrals

7. Some applications of the calculus of variations

8. Some theorems concerning bilinear and multilinear forms

9. Hilbert's inequality and its analogues and extensions

10. Rearrangements