Customer Reviews

Introduction to Inequalities

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This text provides an introduction to the inequalities that form the basis of real analysis, the theoretical foundation for calculus. The authors' treatment requires careful reading since there are many details to check in the derivations of the inequalities and the examples in which those inequalities are applied. In places, I found myself writing annotations in the margins where I found the presentation somewhat incomplete. Checking the details in the authors' exposition and completing the exercises, for which there are answers and hints in the back of the book, is essential for understanding the material.

The text begins with an axiomatic introduction to inequalities. The authors then prove some basic properties of inequalities. The subsequent chapter on absolute value discusses several ways in which absolute value can be interpreted. The most important chapter in the text is one in which some of the most important inequalities in real analysis are derived. In the final chapters of the text, these inequalities are applied to optimization problems and the definition of distance.

The derivations are fascinating, if somewhat ingenious. The authors show the geometric basis of some of the inequalities, a topic the reader can explore further by reading Geometric Inequalities (New Mathematical Library) by Nicholas D. Kazarinoff. Also, the authors show how inequalities can be used to solve problems for which most readers will have been taught quite different methods of solution.

The exercises range from basic computations to proofs for which some ingenuity is required. I wish there were more exercises in the latter chapters of the text to help put the inequalities derived there in context.

The text is a rewarding look at a critical topic in higher mathematics.

It is a very easy book to read. I read it over a six hour flight. It starts with the very basics and takes one through to the most important of inequalities in Mathematics. While inequalities themselves can seem quite uninteresting, this book makes them interesting by focussing on the methodology used to arrive at them and the interesting results that they yield. The book is written in an easy converstational style, and intends to impart the reader not only with the knowledge of some basic inequalities but the authors also succeed in sharing some of the charm and fascination that they hold for pure Mathematics.

I recommend it highly.

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.

This is a very good and well written introductory math book, presented at the high-school or the early college level. The highest prerequisite in terms of math background knowledge is a good familiarity with algebra. Many introductory math texts suffer from the lack of clarity or from relaying too much on the background that is not covered in the book, but this slim volume is fairly self-contained and covers all the material that is used in the problems. The problems are interesting and well formulated. The solutions at teh end of the book are informative, but could have been longer. They are more of a hints than full-fledged solutions, but for the most part this is more than sufficient. The book is useful for anyone who wants to learn about the most important inequalities for the first time or brush up on his/her knowledge. It is also a good exercise book for any math enthusiast.