This is an interesting hodge-podge. It has reasonable coverage of the most-used mathematical inequalities, along with a wide variety of applications, mostly in mechanics and electrical engineering. It is primarily an engineering book, although mathematically rigorous. This second edition adds a brief chapter on interval analysis, that is useful but is not integrated with the rest of the book.

The exposition is choppy and seems to wander around with no real sense of direction. This is in part because about a quarter of the book is Chapter 5 (Applications), a miscellaneous collection with no obvious organizing principle. In the book’s defense, the subject of inequalities is so fragmented that it is hard to provide a clear narrative through it (although Steele’s wonderful *The Cauchy-Schwarz Master Class* does a good job).

One weakness (from a scholar’s perspective) is that when this book cites a book, it doesn’t give the page or section number where the result is found. My favorite bad example is on p. 116, where we are referred to Gradshteyn & Ryzhik’s 1171-page *Table of Integrals, Series, and Products*, without any further direction, for a fact about Legendre polynomials.

Another weak spot is the use of numerical methods for estimates. The book does some numerical examples to demonstrate the tightness of the inequalities. But it skirts the question of when, in applied problems, we should get a good numerical estimate for the quantity (often easy) versus proving analytic upper and lower bounds for the quantity (often hard).

I wasn’t very happy with the exercises, even though they are numerous and have good hints in the back. They are nearly all drill, where one is to apply the general theorems in the text to prove particular inequalities. They test your understanding of the material, but don’t really support the mission of using inequalities in context.

Bottom line: A reasonable introduction to inequalities, but mostly valuable for the variety of applications.

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.