Convex functions with respect to a mean and a characterization of quasi-arithmetic means.

Abstract

Let $M:(0,\infty)^{2}\rightarrow(0,\infty)$ be a homogeneous strict mean such that the function $h:=M(\cdot,1)$ is twice differentiable and $0\neq h^{\prime}(1)\neq1$. It is shown that if there exists an $M$-affine function, continuous at a point which is neither constant nor linear, then $M$ must be a weighted power mean. Moreover the homogeneity condition of $M$ can be replaced by $M$-convexity of two suitably chosen linear functions. With the aid of iteration groups, some generalizations characterizing the weighted quasi-arithmetic means are given. A geometrical aspect of these results is discussed.