Inequalities: Theorems, Techniques and Selected Problems

Front Matter....Pages I-X
Basic (Elementary) Inequalities and Their Application....Pages 1-7
Inequalities Between Means (with Two and Three Variables)....Pages 9-18
Geometric (Triangle) Inequalities....Pages 19-25
Bernoulli’s Inequality, the Cauchy–Schwarz Inequality, Chebishev’s Inequality, Surányi’s Inequality....Pages 27-47
Inequalities Between Means (General Case)....Pages 49-60
The Rearrangement Inequality....Pages 61-67
Convexity, Jensen’s Inequality....Pages 69-77
Trigonometric Substitutions and Their Application for Proving Algebraic Inequalities....Pages 79-94
Hölder’s Inequality, Minkowski’s Inequality and Their Variants....Pages 95-105
Generalizations of the Cauchy–Schwarz Inequality, Chebishev’s Inequality and the Mean Inequalities....Pages 107-116
Newton’s Inequality, Maclaurin’s Inequality....Pages 117-119
Schur’s Inequality, Muirhead’s Inequality and Karamata’s Inequality....Pages 121-132
Two Theorems from Differential Calculus, and Their Applications for Proving Inequalities....Pages 133-136
One Method of Proving Symmetric Inequalities with Three Variables....Pages 137-145
Method for Proving Symmetric Inequalities with Three Variables Defined on the Set of Real Numbers....Pages 147-153
Abstract Concreteness Method (ABC Method)....Pages 155-160
Sum of Squares (SOS Method)....Pages 161-167
Strong Mixing Variables Method (SMV Theorem)....Pages 169-175
Method of Lagrange Multipliers....Pages 177-182
Problems....Pages 183-216
Solutions....Pages 217-433
Back Matter....Pages 435-444