Coinciding mean of the two symmetries on the set of mean functions

TL;DR

This paper investigates two different symmetry definitions on the set of mean functions, showing their equivalence under certain conditions, and introduces a new class of means interpolating harmonic, geometric, and arithmetic means.

Contribution

It provides an answer to an open question about the equivalence of symmetry mappings on mean functions and introduces a new class of interpolating means.

Findings

The two symmetry definitions coincide under specific conditions.

A new class of means interpolating harmonic, geometric, and arithmetic means was discovered.

Asymptotic expansion techniques are effective in analyzing properties of mean symmetries.

Abstract

On the set $\mathcal M$ of mean functions the symmetric mean of $M$ with respect to mean $M_0$ can be defined in several ways. The first one is related to the group structure on $\mathcal M$ and the second one is defined trough Gauss' functional equation. In this paper we provide an answer to the open question formulated by B.\ Farhi about the matching of these two different mappings called symmetries on the set of mean functions. Using techniques of asymptotic expansions developed by T.\ Buri\'c, N.\ Elezovi\'c and L.\ Mihokovi\'c (Vuk\v si\'c) we discuss some properties of such symmetries trough connection with asymptotic expansions of means involved. As a result of coefficient comparison, new class of means was discovered which interpolates between harmonic, geometric and arithmetic mean.