Characterizations of the equality of two-variable generalized quasiarithmetic means

TL;DR

This paper characterizes when two-variable generalized quasiarithmetic means are equal, extending known results about Gini, Stolarsky, and power means through high-order differentiability conditions.

Contribution

It generalizes the characterization of mean equality by showing the intersection of Gini and Stolarsky classes equals quasiarithmetic means under smoothness assumptions.

Findings

The intersection of Gini and Stolarsky means equals quasiarithmetic means.

High-order differentiability is key to the characterization.

Extends classical results to generalized Bajraktarević and Cauchy means.

Abstract

This paper is motivated by an astonishing result of H. Alzer and S. Ruscheweyh published in 2001 in the Proc. Amer. Math. Soc., 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 Bajraktarevi\'c 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.