Algebraic Inequalities

This is a problem book that has a vast collection of particular inequalities. The term “algebraic inequalities” is not defined in the book, but it appears to mean inequalities involving algebraic formulas rather than geometrical quantities. (The authors have a companion book, Geometric Inequalities: Methods of Proving, that is intended to precede this book.) There is some calculus involved, so the methods are not strictly algebraic. Most of it involves high-school level mathematics, and seems partly aimed at the International Mathematical Olympiads. Each chapter contains a long list of inequalities to prove, along with complete proofs, following by another long list of “Problems for Independent Study” that do not have solutions given.

Most chapters deal with a particular method of proof. For example, the first chapter, titled “Basic Inequalities and Their Applications”, is about inequalities that can be proved by subtracting one side of the inequality from the other and manipulating the difference algebraically to get an expression that is clearly non-negative (for example, because it is a perfect square). A simple example of this (actually the second exercise in this chapter) is the simplest form of the Arithmetic Mean–Geometric Mean inequality, \((a+b)/2 \ge \sqrt{ab}\), which is proved by observing that the difference in the two sides is \((a + b)/2 - \sqrt{ab} = (\sqrt{a} - \sqrt{b})^2/2 \ge 0\). Chapter 3 is examples of applying the Arithmetic Mean–Geometric Mean–Harmonic Mean–Quadratic Mean inequalities to prove particular inequalities, and Chapter 4 is examples of the Cauchy–Bunyakovsky–Schwarz inequality. A few chapters do not follow this pattern, but instead gather problems of the same type that are solved by a variety of methods. For example, Chapter 12 is asymptotics of sequences defined by a recurrence, and Chapter 13 is inequalities used in number theory. Chapter 14 is “Miscellaneous Inequalities” and really is miscellaneous — there’s no apparent grouping or pattern to the inequalities or their proofs. There’s also an apparently unrelated Appendix on “Power Sums Triangle”, which is about how to work out the formula for the power sums \(\sum_{i=1}^n i^k\). In general the inequalities to be proved are very specific inequalities rather than generally-applicable inequalities. For example, on p. 6 we are to prove \(3(a + a^2 + a^4) \ge (1 + a + a^2)^2\).The authors say (p. vii) “Most of the inequalities in this book were created by the authors.”

Very Good Feature: brief biographies of the mathematicians who are mentioned. Very Bad Feature: no index.

The big weakness of the book is that the narrative portion is very skimpy: the book claims to teach methods of proving inequalities, but it rarely explains what the methods are. For example, Chapter 2 is on “Sturm’s Method” but does not explicitly explain this method. It appears to be a method for proving multi-variable inequalities when equality occurs when all variables are equal; when not all variables are equal, we adjust their values to make them “more equal” in a way that leaves one side of the inequality unchanged while moving the other side closer to the first side. This is one of the well-known proofs of the Arithmetic Mean–Geometric Mean inequality (in fact due to Sturm), and this is the first problem in the chapter. But this Sturm method is never described explicitly. For another example, Chapter 5 is about changes of variable, but many of the changes seem magical and there’s no indication of how we would have thought of them.

 Bottom line: a good source of inequality problems, but I think students need some extra direction to understand that there are actually general methods being used. Good supplemental books are Kazarinoff’s Analytic Inequalities (elementary but thorough, with lots of examples) and Steele’s The Cauchy–Schwarz Master Class (very strong on methods).

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