An Introduction to Inequalities

Many of formulas and relations in mathematics contain one, or more than one, of the notations < = >. Comparing things with other things is what usually we do, in mathematics or in other branches of science which deal with mathematical tools, and we describe the results, when they contain at least one of < or >, as an inequality. Because of the importance of inequalities, particularly when making required approximations, several mathematicians attempted systematic studies on them at various levels. The book under review is such a book, aimed at high school students.

The series in which the book belongs (the Anneli Lax New Mathematical Library) is directed at high school students. The book wants to enable them:

A. To understand fundamental materials and principles of the theory,
B. To work with inequalities in standard and intelligent manner, and
C. To understand the primary research lines in this area.

The book consists of six chapters: Chapters 1–3 focus on axiomatics and fundamental concepts and tools. In Chapter 4 various classical inequalities of analysis are studied. These include the AGM inequality (the relationship between the arithmetic mean and geometric mean), the Cauchy, Hölder, and Minkowski inequalities, and the triangle inequality. Chapter 5 gives some applications to geometric maximum and minimum problems, and Chapter 6 considers some special distance functions, providing an introduction to Euclidean and Non-Euclidean norms. Each chapter contains some exercises with solutions at the end of book.

The book is classical, but still it has kept its freshness and still seems well-written and well-motivated. This book will also be very useful for teachers, both for its content and as a model of mathematical writing.

Finally, I should mention that the same authors have published another book with the title Inequalities. That book is at a higher level and should not be confused with the book under review.

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics (Analytic Number Theory) in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I in Talence, France, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.