On a preorder relation for Schur-convex functions and a majorization inequality for their gradients and divergences

Abstract

In this paper, a preorder relation for Schur-convex functions on $$ {\mathbb {R}}^{n}$$ is introduced. A majorization statement is shown for the gradients and divergences of two Gateaux differentiable Schur-convex functions, provided that the difference of the involved functions is also Schur-convex. This implies the monotonicity of a related operator with respect to the used preorder and the classical majorization preorder on $$ {\mathbb {R}}^{n}$$ . Special cases of the main result are also studied. In particular, applications are given for strongly convex functions. Some comparisons of variances are presented for uniform distribution.