Inequalities

- Basic (elementary) inequalities and their application

- Inequalities between means, (with two and three variables)

- Geometric (triangle) inequalities

- Bernoulli's inequality, the Cauchy-Schwarz inequality, Chebishev's inequality, Surányi's inequality

- Inequalities between means (general case)

- Points of incidence in applications of the AM-GM inequality

- The rearrangement inequality

- Convexity, Jensen's inequality

- Trigonometric substitutions and their application for proving algebraic inequalities

- The most usual forms of trigonometric substitutions

- Characteristic examples, using trigonometric substitutions

- Hölder's inequality, Minkowski's inequality and their generalizations

- Generalizations of the Cauchy-Schwarz inequality, Chebishev's inequality and the mean inequalities

- Newton's inequality, Maclaurin's inequality

- Schur's inequality, Muirhead's inequality

- Two theorems from differential calculus, and their applications for proving inequalities

- One method of proving symmetric inequalities with three variables

- Method for proving symmetric inequalities with three variables defined on set of real numbersapplications of the AM-GM inequality

- The rearrangement inequality

- Convexity, Jensen's inequality

- Trigonometric substitutions and their application for proving algebraic inequalities

- The most usual forms of trigonometric substitutions

- Characteristic examples, using trigonometric substitutions

- Hölder's inequality, Minkowski's inequality and their generalizations

- Generalizations of the Cauchy-Schwarz inequality, Chebishev's inequality and the mean inequalities

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Dipl. Math. Zdravko Cvetkovski, University of Macedonia R. Macedonia, Informatics Department.

From the reviews:"This volume collects problems of various degrees of difficulty in the field of elementary inequalities. This book is intended as a valuable source for the training of high-school students in view of mathematical Olympiads. School teachers will also gain benefit from this book. ... It would be a particularly valuable resource for those who participate in mathematics competitions at the high school or college level. ... this book is a 'must have' for a university's library, and I recommend it highly to its 'ideal audience'." (Teodora-Liliana Radulescu, Zentralblatt MATH, Vol. 1233, 2012)

From the reviews:

"The book is aimed at the more advanced students about to enter university and is particularly useful to candidates, and their trainers, for mathematical competitions such as the Olympiads. The themes and methods are introduced in 19 chapters of the book, which include exercises and their solutions." (Peter Shiu, The Mathematical Gazette, Vol. 98 (541), March, 2014)"This volume collects problems of various degrees of difficulty in the field of elementary inequalities. This book is intended as a valuable source for the training of high-school students in view of mathematical Olympiads. School teachers will also gain benefit from this book. ... It would be a particularly valuable resource for those who participate in mathematics competitions at the high school or college level. ... this book is a 'must have' for a university's library, and I recommend it highly to its 'ideal audience'." (Teodora-Liliana Radulescu, Zentralblatt MATH, Vol. 1233, 2012)