My Favorite NML Book: An Introduction to Inequalities

Dr. Timothy G. Feeman, a mathematics professor at Villanova University, discusses his favorite Anneli Lax New Mathematical Library book. "Among the slender, inviting volumes in the Anneli Lax New Mathematical Library (NML) series, one that appeals to my analytical bent and continues to provide food for thought is An Introduction to Inequalities, by Edwin Beckenbach and Richard Bellman. After a full 40 pages of discussion of the meaning of the “greater than” symbol, the order relation on the real number system, and everything you wish your Advanced Calculus students knew about absolute values comes the first main theme of the book: the inequality of the arithmetic and geometric means. The treatment builds inevitably and meticulously from the AMGM inequality for two numbers to the general version for n numbers, culminating in the demonstration of Young’s inequality. The Cauchy inequality is the second theme, and the two themes together lead us to the discovery of the three big inequalities for metric spaces: Hӧlder, Minkowski, and the triangle inequality. An iteration of the AMGM inequality due to Gauss leads us to the verge of the study of elliptic functions. In keeping with the nature of the NML series, all of this is done without any mention of Calculus. Geometric interpretations of the inequalities get their full due attention. The reward for our efforts comes in the treatment of max/min problems, still without Calculus of course, in Chapter 5. The problem of “the wealthy football player”, who wishes to have his rectangular coffin interred in an ellipsoidal crypt, is a real treat. The book wraps up with a wonderful little introduction to metric spaces, including a proof that the unit disc in a metric space must be convex and symmetric about the origin. Inequalities are the bread and butter of analysis; this book is like wholesome multi-grain bread spread with the freshest creamery butter. Dig in!" To purchase An Introduction to Inequalities, visit the MAA eBooks Store.