On the $\sigma$-balancing property of multivariate generalized quasi-arithmetic means

TL;DR

This paper characterizes the $\sigma$-balancing property in multivariate generalized quasi-arithmetic means, extending previous results by relaxing conditions on generating functions and exploring the property’s implications.

Contribution

It extends the characterization of the $\sigma$-balancing property to multivariate means under minimal assumptions, broadening the understanding of quasi-arithmetic means.

Findings

Characterization of $\sigma$-balancing property in multivariate means

Relaxation of differentiability conditions on generating functions

Extension of previous univariate results to multivariate case

Abstract

The aim of this paper is to characterize the so-called $\sigma$-balancing property in the class of generalized quasi-arithmetic means. In general, the question is whether those elements of a given family of means that possess this property are quasi-arithmetic. The first result in the latter direction is due to G. Aumann who showed that a balanced complex mean is necessariliy quasi-arithmetic provided that it is analytic. Then Aumann characterized quasi-arithmetic means among Cauchy means in terms of the balancing property. These results date back to the 1930s. In 2015, Lucio R. Berrone, generalizing balancedness, concluded that a mean having that more general property is quasi-arithmetic if it is symmetric, strict and continuously differentiable. A common feature of these results is that they assume a certain order of differentiability of the mean whether or not it is a natural condition. In 2020, the balancing property was characterized in the family of generalized quasi-arithmetic means of two variables under only natural conditions, namely continuity and strict monotonicity of their generating functions. Here we extend the corresponding result for multivariate generalized quasi-arithmetic means by relaxing the conditions on the generating functions and considering the more general $\sigma$-balancing property.