In the Little Mathematics Library series we now come to Inequalities by P. P. Korovkin.

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. The book contains 63 problems, 35 of which are provided with detailed solutions, composing thus its main subject, and 28 others are given in Sections 1.1 and 2.1, 2.3, 2.4 as exercises for individual training. At the end of the book the reader will find the solutions to the given exercises.

The book was translated the Russian by *Sergei Vrubel* and was first published by Mir in 1975.

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Contents

Preface 6

Chapter 1. Inequalities 7

1.1. The Whole Part of a Number 7

1.2. The Arithmetic Mean and the Geometric Mean 12

1.3. The Number 19

1.4. The Bernoulli Inequality 23

1.5. The Mean Power of Numbers 27

Chapter 2. Uses of Inequalities 32

2.1. The Greatest and the Least Function Values 32

2.2. The Holder Inequality 40

2.3. The Use of Inequalities for Calculation of Limits 43

2.4. The Use of Inequalities for Approximate Calculation of Quantities 49

Solutions to Exercises 58

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