This is a somewhat specialized monograph on inequalities, focusing on integral inequalities and the use of convexity. The subtitle “A Perspective” means a personal perspective, that is, the book deals with inequalities that the authors have worked on and are interested in, and not with inequalities in general.

The work is deep rather than wide: it usually starts with a familiar inequality and then proves generalizations, analogs, and variants. One of the most interesting aspects is many instances of “reverses” (some would call them “converses”). This is where you have a familiar inequality, *A* ≤ *B*, and a reverse is a result *B* ≤ *p A* proved under additional hypotheses and with a parameter *p* that depends on some aspects of the quantities involved.

This book is strictly a reference, with no exercises. One big weakness of the book is that all inequalities are treated in isolation: we don’t know why they were invented or what they might be good for. For example, there is a “Bombieri’s inequality,” which is the key ingredient in the large sieve in number theory and leads to a number of other interesting inequalities, but we are not told any of this.

One peculiarity of the book is that every inequality seems to be named after somebody (as in Bombieri’s inequality). These attributions are probably accurate, but are not standard. Often the inequalities are not referenced back to the original papers, but only to modern treatments by the book’s authors, so it’s not easy to evaluate these namings. There is also some non-standard terminology. For example, on p. 103 we are introduced to Hadamard’s inferior sum and Hadamard’s superior sum, which any calculus teacher will recognize as the midpoint and trapezoidal rules for integration.

Bottom line: a useful book if you are interested in its specific subject matter, but not a good book to start learning about inequalities. The best all-around inequality book is Steele’s The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. It not only has very broad coverage of inequalities, but will teach you how to prove your own inequalities.

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.