Convexity of power functions with respect to symmetric homogeneous means

Abstract

It is well known that the power function $x \mapsto {x^p}$ on $\mathbb{R}_{ + }^{*}$ is strictly Jensen-convex if p ²−p>0, strictly Jensen-concave if p ²−p<0, Jensen-convex and Jensen-concave if p ²−p=0. These Jensen type properties are based upon the arithmetic mean A : $A:\mathbb{R}_{ + }^{*} \times \mathbb{R}_{ + }^{*} \to \mathbb{R}_{ + }^{*}$. It is the purpose of this paper to investigate the convexity/concavity classification of the power functions for symmetric homogeneous means on $\mathbb{R}_{ + }^{*}$ other than A. In Section 3, a convexity/concavity criterion is presented, and in Section 4 this is applied to the families of Stolarsky means and Gini means (both containing A) as well as to weighted geometric means.