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Secrets in Inequalities - basic inequalities

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Titlu: Secrets in Inequalities - basic inequalities

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It is true that there are very many books on inequalities and you have all the right to be bored and tired of them. But we tell you that this is not the case with this one. Just read the proof of Nesbitt's Inequality in the very beginning of the material, and you will understand exactly what we mean.

Every topic is described through various and numerous examples taken from many sources, especially from math contests around the world, from recent contests and recent books, of from (more or less) specialized sites on the Internet, which makes the book very lively and interesting to read for those who are involved in such activities, students and teachers from all over the world.

Don't let the problems overwhelm you, though they are quite impressive problems, study applications of the first five basic inequalities mentioned above, plus the Abel formula, symmetric inequalities and the derivative method. Now relax with the AM-GM inequality - the foundational brick of inequalities.


Abbreviations and Notations

I The basic Inequalities
1 AM-GM nequality
1.1 AM-GM Inequality and Applications
1.2 The Cauchy Reverse Technique

2 Cauchy-Schwarz and Holder inequalities
2.1 Cauchy-Schwarz inequality and Applications
2.2 Holder Inequality

3 Chebyshev Inequality
3.1 Chebyshev Inequality and Applications
3.2 The Chebyshev Associate Technique

4 Inequalities with Convex Functions
4.1 Convex functions and Jensen inequality
4.2 Convex Functions and Inequalities with Variables Restricted to an Interval

5 Abel Formula and Rearrangement Inequality
5.1 Abel formula
5.2 Rearrangement Inequality

6 The Method of Balanced Coefficients
6.1 Balancing coefficients by AM-GM inequality
6.2 Balancing coefficients by Cauchy-Schwarz and Holder inequalities

7 Derivative and Applications
7.1 Derivative of one-variable functions
7.2 Derivative of n-Variable Functions

8 A note on symmetric inequalities
8.1 Getting started
8.2 Primary symmetric polynomials
8.3 Normalization skill
8.4 Symmetric separation

9 Problems and Solutions


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