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Secrets in Inequalities - advanced inequalities

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Titlu: Secrets in Inequalities - advanced inequalities

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As you have seen in "Secrets in Inequalities. Volume 1" and also from the title, this volume is about advanced inequalities. The main idea which let us start this project was a book about inequalities. There are many, we have known but we want them organized in a different way. This book will be very well understood by those who have already read Volume 1, and more than this it will be the continuation of the nominated book just like the second part of a trip, a trip in the world of inequalities.

Remarkably, the method of using classical inequalities is left behind the first four methods, because we realize that you can flexibly use this method only after you have already developed a larger horizon in the field of inequalities; and this is true: a solid background in a field is usually needed to find the simplest and the most natural solution to a problem. You will find here good modern approaches to prove inequalities: the mixing variables method (in its general and special forms), the method of analyzing squares, the contradiction method, the method of induction (and again, the above mentioned method of using classical inequalities). As you are already used from previous parts, you have lots of beautiful and hard problems to exercise your skills in using all these methods.


Abbreviations and Notations

1 Analyzing Squares Method
1.1 Getting Started
1.2 The Standard Representation
1.3 Symmetric Inequalities and SOS Method
1.4 Cyclic Inequalities and SOS Method
1.5 The Exercises

2 Mixing Variable Method
2.1 Getting Started
2.2 Typical Applications
2.3 Stronger Mixing Variable Method
2.4 Mixing All Variables Method
2.5 Reference: The Global Derivative
2.6 The Exercises

3 Contradiction Method
3.1 Getting Started
3.2 Step by Step
3.3 Typical Applications
3.4 The Exercises

4 General Induction Method
4.1 Getting Started
4.2 Step by Step
4.3 Typical Applications
4.4 The Exercises

5 Method of using Classical Inequalities
5.1 Problem series A. Ingenious Applications
5.2 Problem series B. Clever Separations
5.3 Problem series C. Incredible Identities
5.4 Problem series D. The Non-homogeneous

6 Collected Articles on Inequalities

7 The Readers' Contributions
Contributions 1. Undefined Mixing Variable Method
7.1 Introduction
7.2 Starting with a simple idea
7.3 The Undefined Mixing Variable method
7.4 Applications for Inequalities of Degree 3
Contributions 2. SOS-Schur Method
7.5 Introduction
7.6 The SOS-Schur Representation
7.7 Applications of SOS-Schur Method
7.8 A special technique
7.9 Another Examples


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