Symmetrization, convexity and applications

Abstract

Based on permutation enumeration of the symmetric group and ‘generalized’ barycentric coordinates on arbitrary convex polytope, we develop a technique to obtain symmetrization procedures for functions that provide a unified framework to derive new Hermite–Hadamard type inequalities. We also present applications of our results to the Wright-convex functions with special emphasis on their key role in convexity. In one dimension, we obtain (up to a positive multiplicative constant) a method of symmetrization recently introduced by Dragomir (2014) [3], and also by El Farissi et al. (2012/2013) [4]. So our approach can be seen as a multivariate generalization of their method.