Scales of quasi-arithmetic means determined by invariance property

TL;DR

This paper characterizes a family of quasi-arithmetic means that form a scale based on an invariance property, extending the classical understanding of power means and their uniqueness under homogeneity.

Contribution

It introduces an invariance-based axiom that generalizes the scale property of power means to a broader class of quasi-arithmetic means.

Findings

The family of power means forms a scale with respect to the parameter t.

Replacing homogeneity with invariance yields a new scale of quasi-arithmetic means.

The results extend classical theorems about the uniqueness of power means under different axioms.

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.