Bulletin of the
Korean Mathematical Society BKMS

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

Keywords: Schur convexity, Schur power convexity, Gini means

MSC numbers: Primary 26B25, 26E60; Secondary 26D15