The present booklet contains some particularly interesting inequalities playing an important role in various sections of higher mathematics. These inequalities are used for finding the greatest and least values as well as for calculating the limits. The booklet contains 63 problems and most of them are provided with detailed solutions.

The book is intended for students of senior classes of secondary schools.

From the preface …

In the mathematics course of secondary schools students get acquainted with the properties of inequalities and methods of their solution in elementary cases (inequalities of the first and second degree).

In this booklet the author did not pursue the aim of presenting the basic properties of inequalities and made an attempt only to familiarize students of senior classes with some particularly remarkable inequalities playing an important role in various sections of higher mathematics and with their use for finding the greatest and the least values of quantities and for calculating some limits.

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The solution of some difficult problems carried out individually will undoubtedly do the reader more good than the solution of a large number of simpler ones.

For this reason we strongly recommend the readers to perform their own solutions before referring to the solutions given by the author at the end of the book. However, one should not be disappointed if the obtained results differ from those of the patterns. The author considers it as a positive factor.

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Summary: Outstanding short introduction to inequalities, but better translation available (see review) Rating: 5

Note there are two English language version of this book. There is not a great deal of difference between them, but they each have their strength and weaknesses (see review below).

This short paperback pamphlet by P. P. Korovkin, in either version, is one of the best and shortest introductions to inequalities and their use. It should be accessible to general readers with a solid foundation of high school mathematics.

Here, the concepts material is divided into five chapters of about 10 pages each, followed by a final chapter with exercise solutions. The introductory three chapters present basic principles and the final two chapter (before the solution section) present applications of inequalities to "Maxima and Minima" and to the "Calculation of Some Limits".

This particular edition has some font and translation issues. The text was prepared by typewriter, and is a bit difficult to read. The translation by H. Moss has the occasional awkward sentence. However, the layout is excellent with formulas nicely separated from text.

There is a better translation of this work printed by the Russian publisher MIR, available in their "Little Mathematics Library". That slightly smaller-formatted edition is also set in a more readable font. However, that pamphlet's layout is not as satisfactory, with formulas less well separated from the supporting text.

This edition has a slightly sturdier cover than the MIR edition. It's more like a Dover paperback, probably familiar to most mathematics readers. Either edition provides an outstanding introduction; the MIR edition has the better translation and font, and this edition a better layout of formulas.