SCHUR POWER CONVEXITY OF GINI MEANS

Abstract

In this paper, the Schur convexity is generalized to Schur <TEX>$f$</TEX>-convexity, which contains the Schur geometrical convexity, harmonic convexity and so on. When <TEX>$f$</TEX> : <TEX>${\mathbb{R}}_+{\rightarrow}{\mathbb{R}}$</TEX> is defined as <TEX>$f(x)=(x^m-1)/m$</TEX> if <TEX>$m{\neq}0$</TEX> and <TEX>$f(x)$</TEX> = ln <TEX>$x$</TEX> if <TEX>$m=0$</TEX>, the necessary and sufficient conditions for <TEX>$f$</TEX>-convexity (is called Schur <TEX>$m$</TEX>-power convexity) of Gini means are given, which generalize and unify certain known results.