Inequalities for Convex and Concave Functions and a New Concept of Majorization Intended for Two Pairs of Vectors

Abstract

In this paper, some inequalities for convex and concave functions are demonstrated. A new concept of majorization is applied for two pairs of vectors. Instead of the standard approach based on one doubly stochastic matrix, here two matrices are employed: one is both row stochastic and column substochastic, while the other is both column stochastic and row superstochastic. An inequality of Hardy–Littlewood–Polya–Karamata type is derived for a row stochastic–column substochastic matrix.