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

Janusz Matkowski
Source: Real Anal. Exchange Volume 29, Number 1 (2003), 229-246.

#### 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.

First Page:
Primary Subjects: 26A51, 26E60, 39B22
Secondary Subjects: 39B12
Full-text: Open access
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.rae/1149860188
Mathematical Reviews number (MathSciNet): MR2061307
Zentralblatt MATH identifier: 1067.26008