Inequalities : Theorems, Techniques and Selected Problems

This Work Is About Inequalities Which Play An Important Role In Mathematical Olympiads. It Contains 175 Solved Problems In The Form Of Exercises And, In Addition, 310 Solved Problems. The Book Also Covers The Theoretical Background Of The Most Important Theorems And Techniques Required For Solving Inequalities. It Is Written For All Middle And High-school Students, As Well As For Graduate And Undergraduate Students. School Teachers And Trainers For Mathematical Competitions Will Also Gain Benefit From This Book. 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 Numbers -- Abstract Concreteness Method (abc Method) -- Sum Of Squares (s.o.s - Method) -- Strong Mixing Variables Method (s.m.v Theorem) -- Lagrange Multipliers Method. Zdravko Cvetkovski. Includes Bibliographical References (p. 443-444) And Index. 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