An Introduction to Inequalities is an unexpectedly delightful book. Relatively brief, only 129 pages, this publication of The Mathematical Association of America, requires no more than basic high school mathematics. Nonetheless, I am convinced that Edwin Beckenbach's and Richard Bellman's systematic study of inequalities would interest most students in an early calculus course.
The first two short chapters establish an axiomatic framework for the algebra of inequalities that should be familiar to most readers. Even so, it best not to skip the nine problems at the end of chapter 2 as the results will play important roles in later chapters.
Chapter 3, Absolute Value, offers an interesting look at what I had generally considered to be a prosaic topic. Beginning with a straight-forward definition, the authors derive some half-dozen expressions for the absolute value. This discussion leads to the triangle inequality (one-dimensional case).
The next chapter, The Classical Inequalities, is a gem. (Many readers could probably go directly to chapter 4, but the first three chapters are quick reading in any case.)
Some classical inequalities were familiar, like the arithmetic mean - geometric mean inequality and the Cauchy inequality (two-dimensional version). But others like the n-dimensional version of the Cauchy inequality (along with the Cauchy-Lagrange identity), the Holder inequality, and the Minkowski inequality were new to me. What I found most surprising was how these classical inequalities were so interrelated, and how some can be considered generalizations of others. Beckenbach and Bellman introduce clever substitutions to transform one inequality expression into another.
Side note: I was intrigued with a trapezoidal representation (a single drawing) that geometrically related the inequalities for the arithmetic mean, the geometric mean, the harmonic mean, and the root-mean square. Developing geometry proofs for arithmetic and geometric means was not difficult, but I needed the helpful hints in the answer section for the harmonic mean and root-mean square.
In Chapter Five the authors solve a range of maximization and minimization problems using inequalities rather than the techniques of calculus. A final short chapter enumerates some of the properties that extend the familiar notion of Euclidian distance to various examples of non-Euclidian distance functions.
I have now ordered another MAA publication, Geometric Inequalities by Nicholas D. Kazarinoff, which is not only a good text in itself, but makes a good folow-up as it often references Beckenbach and Bellman. I have also purchased a Dover reprint titled Analytic Inequalities by Nicholas D. Kazarinoff. More terse and more technical, this second text targets undergraduate mathematics majors.
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