# Characterization of the equality of Cauchy means to quasiarithmetic means

## TL;DR

This paper establishes six necessary and sufficient conditions under which a Cauchy mean coincides with a quasiarithmetic mean, linking the equality to geometric properties of the generating functions.

## Contribution

It provides a complete characterization of when Cauchy means are equivalent to quasiarithmetic means through geometric and regularity conditions.

## Key findings

- Six necessary and sufficient conditions identified
- Range of generating functions linked to conic sections
- Characterization under various regularity assumptions

## Abstract

The main result of this paper provides six necessary and sufficient conditions under various regularity assumptions for a so-called Cauchy mean to be identical to a two-variable quasiarithmetic mean. One of these conditions says that a Cauchy mean is quasiarithmetic if and only if the range of its generating functions is covered by a nondegenerate conic section.