Cvetkovski Z. Inequalities: Theorems, Techniques and Selected Problems

Springer-Verlag Berlin, 2012, ISBN: 3642237916, 444 pagesThe book is designed and intended for all students who wish to expand their knowledge related to the theory of inequalities and those fascinated by this field. The book could be of great benefit to all regular high school teachers and trainers involved in preparing students for national and international mathematical competitions as well.
As an integral part of the book, following the development of the theory in each section, solved examples have been included—a total of 175 in number — all intended for the student to acquire skills for practical application of previously adopted theory. Also should emphasize that as a final part of the book an extensive collection of 310 high quality solved problems has been included, in which various types of inequalities are developed. Some of them are mine, while the others represent inequalities assigned as tasks in national competitions and national olympiads as well as problems given in team selection tests for international competitions from different countries.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 Variants
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 and Karamata’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 the Set of Real Numbers
Abstract Concreteness Method (ABC Method)
ABC Theorem
Sum of Squares (SOS Method)
Strong Mixing Variables Method (SMV Theorem)
Method of Lagrange Multipliers
Problems
Solutions