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Inequalities. An approach through problems. (English) Zbl 1175.26002

Texts and Readings in Mathematics 49. New Delhi: Hindustan Book Agency (ISBN 978-81-85931-88-3/hbk). viii, 392 p. (2009).
This book gives us a comprehensive presentation of inequalities and their use in mathematical contests problems. The book is divided into six chapters: Some basic inequalities, Techniques for proving inequalities, Geometric inequalities, Applications involving inequalities, Problems on inequalities, Solutions to problems.
In the first chapter, classical inequalities are given. There we can find the AM-GM inequality, the Cauchy-Schwarz inequality, Jensen’s, Chebyshev’s, Hölder’s and Minkowski’s inequalities, the rearrangement inequality, inequalities for symmetric functions. They are given in a discrete form because integral inequalities are not a part of competition mathematics and, in addition, they are proved. Each classical inequality is accompanied with some related problems. The second chapter describes different methods for proving inequalities such as using induction, using calculus, trigonometric substitutions, majorisation and homogenisation technique etc. Each technique is illustrated with several examples.
In the third chapter, a sequence of geometric inequalities is given (more than 70). Of course, it is not an exhaustive list of geometric inequalities, but those problems illustrate how earlier developed techniques can be used in this field. Examples from the fourth chapter show us how some problems, which are not direct inequality problems, can be solved by inequalities.
Finally, the fifth chapter contains 239 problems, mainly taken from mathematical contests all around the world. Detailed proofs of those problems are given in the last part of book.
This book is recommended for students participating in mathematical competitions and their trainers. There are more than 400 examples and problems all provided with detailed solutions.

MSC:

26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
26D05 Inequalities for trigonometric functions and polynomials
26D07 Inequalities involving other types of functions
26D15 Inequalities for sums, series and integrals
26D20 Other analytical inequalities
00A07 Problem books
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