116 Algebraic Inequalities from the AwesomeMath Year-Round Program

This is a problem book in inequalities, that specializes in the kind of inequalities found in the International Mathematical Olympiads (IMOs). The “algebraic” in the title mostly means that it does not cover geometric inequalities (see the companion book 113 Geometric Inequalities from the AwesomeMath Summer Program), but also that there are no integral inequalities and that the methods are based on (college) algebra, with almost no infinite processes. The book is one of a long series of IMO-related books put out by XYZ Press, most of them co-authored by Titu Andreescu, a prominent figure in the United States IMO training.

The first half of the book covers the big three inequalities (Arithmetic Mean–Geometric Mean, including the more general power means; Cauchy–Schwarz inequality; and Jensen’s inequality), along with a number of slightly lesser-known inequalities, and has many illustrative examples. The second half of the book comprises the 116 problems (with complete solutions), most of which resemble IMO problems.

The book is slanted very much toward the IMOs, but the expository parts of the first half give a good introduction to inequalities, no matter what your interests are. A conspicuous weakness is that there are no graphs or drawings; many inequalities are easier to understand with a graph.

There is unfortunately no bibliography or suggestions for further reading. Two good books at the same level are Beckenbach & Bellman’s An Introduction to Inequalities and Kazarinoff’s Analytic Inequalities (the latter is slightly more advanced, as it uses some calculus). The best all-around book on inequalities is Steele’s The Cauchy–Schwarz Master Class; it is aimed at math researchers, but much of it is elementary enough to use at this level, and it does have very good problems. All three books make good use of graphs and drawings to illustrate the inequalities.

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.