Characterization of generalized quasi-arithmetic means

TL;DR

This paper provides a mathematical characterization of generalized quasi-arithmetic means using Gauss composition and bisymmetry equations, advancing the theoretical understanding of these means.

Contribution

It introduces a novel characterization of generalized quasi-arithmetic means through Gauss composition and bisymmetry, extending existing mathematical frameworks.

Findings

Characterization involves Gauss composition of cyclic mean-type mappings.

Generalized bisymmetry equation is key to the characterization.

Provides a new theoretical framework for understanding these means.

Abstract

In this paper we characterize generalized quasi-arithmetic means, that is means of the form $M(x_1,...,x_n):=(f_1+...+f_n)^{-1}(f_1(x_1)+...+f_n(x_n))$, where $f_1,...,f_n:I\to\mathbb{R}$ are strictly increasing and continuous functions. Our characterization involves the Gauss composition of the cyclic mean-type mapping induced by $M$ and a generalized bisymmetry equation.