This work is about inequalities which play an important role with mathematical olympiads. It contains 150 solved problems given as exercises and, in addition, 308 solved problems. This book also covers the theoretical background of the most important theorems and techniques required to solve inequalities. It is written for all middle and high school students, as well as for graduate andundergraduate students. School teachers and trainers for mathematical competitions will also benefit from the wealth of Algebra-Inequalities. Contains 458 problems on Algebra-Inequalities. Detailed solutions are given. Theoretical background and techniques for proving inequalities are provided. Ideal for participants of national and international mathematical contests. INDICE: '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 themean 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.
- ISBN: 978-3-642-23791-1
- Editorial: Springer Berlin Heidelberg
- Encuadernacion: Rústica
- Páginas: 382
- Fecha Publicación: 31/12/2011
- Nº Volúmenes: 1
- Idioma: Inglés
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- MATEMÁTICAS /
- ÁLGEBRA