Interval-type theorems concerning quasi-arithmetic means

TL;DR

This paper explores the structure of quasi-arithmetic means, introducing interval-type sets based on point-wise orderings and examining how smoothness assumptions of generating functions influence these sets.

Contribution

It provides new examples of interval-type sets for quasi-arithmetic means considering different smoothness conditions of the generating functions.

Findings

Characterization of the partial order of quasi-arithmetic means

Introduction of interval-type sets within this order

Examples of such sets based on smoothness assumptions

Abstract

Family of quasi-arithmetic means has a natural, partial order (point-wise order) $A^{[f]}\le A^{[g]}$ if and only if $A^{[f]}(v)\le A^{[g]}(v)$ for all admissible vectors $v$ ($f,\,g$ and, later, $h$ are continuous and monotone and defined on a common interval). Therefore one can introduce the notion of interval-type sets (sets $\mathcal{I}$ such that whenever $A^{[f]} \le A^{[h]} \le A^{[g]}$ for some $A^{[f]},\,A^{[g]} \in \mathcal{I}$ then $A^{[h]} \in \mathcal{I}$ too). Our aim is to give examples of interval-type sets involving vary smoothness assumptions of generating functions.