Dictionary of Inequalities

Because of natural need to compare quantities together, inequality is one of the most fundamental concepts in mathematics. If we consider the several branches of mathematics, we find in almost all a footprint of inequalities. But the development of a “mathematical theory of inequalities” as a sub-branch of analysis was formally initiated only after the book Inequalities, by G. H. Hardy, J. E. Littlewood and G. Pólya.

Since then, many books and monographs for collecting and systematizing inequalities were published, though the word “inequality” only appeared in the Mathematics Subject Classification of the American Mathematical Society in 1982. Today, inequality theory is an active branch of mathematics, and has at least 6 specialized journals, by my count. These journals and books contain a large number of inequalities, some of have names attached to them: the Arithmetic-Geometric-Harmonic mean inequalities, the Cauchy inequality, the Gram inequality, the Hermite-Hadamard inequality, Hölder’s inequality, etc.

As its title indicates, the book under review is a collection, in a friendly volume, of well-known and less-known inequalities and their possible generalizations and refinements. The book doesn’t include proofs, and to understand most parts of it only undergraduate calculus, analysis and probability is needed. It is organized in 23 chapters. Each chapter consists of several sections headed by the name of inequality under study, enabling readers to access a desired inequality in a very easy and fast way. The book contains a list of URLs, and its bibliography consists of “Basic References”, “Collections and Encyclopedias”, “Books” and a long list of “Papers”, which is very useful for finding the main sources of the results. This book may also be useful for instructors of calculus and basic analysis to find some very meaningful exercises, leading students to borders of research in inequality theory and its applications.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.