Inequalities (eBook)

Dipl. Math. Zdravko Cvetkovski, European University-Skopje, R. Macedonia, Informatics Department.

"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.