INEQUALITIES FOR GENERALIZED CONVEX FUNCTIONS WITH APPLICATIONS, I

Abstract

 The Theory of Inequalities has a majore role in Mathematical Analysis, and in almost all areas of Mathematics, too. In this theory, the convex functions and the generalized convexity plays a special role. The author has published a series of papers with applications of convexity inequalities in various fields of Mathematics. We quote applications in geometry (see e.g. [16], [22]), special functions ([19], [18], [23]); number theory (see many articles collected in the monograph [34]); the theory of means ([24], [25], [31], [33]), etc. The aim of this series of papers (planned to have 4 parts) is to survey the most important ideas and results of the author in the theory of convex inequalities. In the course of this survey, many new results and applications will be obtained. In most cases only the new results will be presented with a proof; the other results will be stated only, with connections and/or applications to known theorems. All material is centered around three most important inequalities, namely: Jensen's inequality, Jensen-Hadamard's (or Hermite-Hadamard's) inequality and Jessen's inequality.