Characterizations of the equality of two-variable generalized quasiarithmetic means

Zsolt Páles,Amr Zakaria

doi:10.1016/j.jmaa.2021.125813

Zsolt Páles, Amr Zakaria

Open Access

https://doi.org/10.1016/j.jmaa.2021.125813

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Abstract

This paper is motivated by an astonishing result of H. Alzer and S. Ruscheweyh published in 2001, which states that the intersection of the classes two-variable Gini means and Stolarsky means is equal to the class of two-variable power means. The two-variable Gini and Stolarsky means form two-parameter classes of means expressed in terms of power functions. They can naturally be generalized in terms of the so-called Bajraktarević and Cauchy means. Our aim is to show that the intersection of these two classes of functional means, under high-order differentiability assumptions, is equal to the class of two-variable quasiarithmetic means.

Highlights

A notion, which subsumes the concept of arithmetic, geometric and harmonic means is the concept of power means
The class of two-variable power means has been extended in numerous ways in the literature
We will recall further important classes of two-variable functional means that extend Holder, Gini and Stolarsky means in a natural way

Summary

Introduction

A notion, which subsumes the concept of arithmetic, geometric and harmonic means is the concept of power means. Power mean; Bajraktarevicmean; Cauchy mean; Gini mean; Stolarsky mean; generalized quasiarithmetic mean; equality problem; functional equation. We will recall further important classes of two-variable functional means that extend Holder, Gini and Stolarsky means in a natural way. These classes are the quasiarithmetic, Bajraktarevic, and Cauchy means. A recent characterization of this equality in terms of eight equivalent conditions has been established in [18, Theorem 15] Another important generalization of quasiarithmetic means was introduced as follows: If f, g : I → R are continuously differentiable functions with g′ ∈ CP(I) and f ′/g′ ∈ CM(I), define Cf,g : I2 → I by f ′ −1 f (x) − f (y). As a consequence of this result, it will follow that the intersection of these two classes of means consists of quasiarithmetic means

Auxiliary results on measures

Auxiliary results on generalized quasiarithmetic means