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	• Introduction to Inequalities (New Mathematical Library) Edwin F. Beckenbach, R. Bellman Mathematical Association of America (MAA), 1975 - 133 pages average customer review:based on 3 reviews view larger image
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I recommend it highly.

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An interesting study of inequalities.

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.

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Enlivens the Study of Inequalities - From Fundamentals to Classical Inequalities to Min - Max Problems

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, and 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 a short text, a Dover reprint, Analytic Inequalities by Nicholas D. Kazarinoff. More terse and more technical, this text targets undergraduate mathematics majors.

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Most people, when they think of mathematics, think first of numbers and equations-this number (x) = that number (y). But professional mathematicians, in dealing with quantities that can be ordered according to their size, often are more interested in unequal magnitudes that areequal. This book provides an introduction to the fascinating world of inequalities, beginning with a systematic discussion of the relation "greater than" and the meaning of "absolute values" of numbers, and ending with descriptions of some unusual geometries. In the course of the book, the reader wil encounter some of the most famous inequalities in mathematics.

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