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

In this paper, a preorder relation for Schur-convex functions on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb {R}}^{n}$$\end{document} 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 Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb {R}}^{n}$$\end{document}. 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.