Scales of quasi-arithmetic means determined by invariance property

Abstract

It is well known that if $\mathcal{P}_t$ denotes a set of power means then the mapping $\mathbb{R} \ni t \mapsto \mathcal{P}_t(v) \in (\min v, \max v)$ is both 1-1 and onto for any non-constant sequence $v = (v_1,\dots,\,v_n)$ of positive numbers. Shortly: the family of power means is a scale. If $I$ is an interval and $f \colon I \rightarrow \mathbb{R}$ is a continuous, strictly monotone function then $f^{-1}(\tfrac{1}{n} \sum f(v_i))$ is a natural generalization of power means, so called quasi-arithmetic mean generated by $f$. A famous folk theorem says that the only homogeneous, quasi-a\-rith\-me\-tic means are power means. We prove that, upon replacing the homogeneity requirement by an invariant-type axiom, one gets a family of quasi-arithmetic means building up a scale, too.