On a functional equation related to two-variable Cauchy means

TL;DR

This paper solves a functional equation involving unknown functions and applies the results to compare two-variable Cauchy and quasi-arithmetic means, extending previous work by assuming only continuity of a key function.

Contribution

It provides a complete description of solutions to a specific functional equation under minimal continuity assumptions and applies these solutions to mean comparison problems.

Findings

Derived explicit solutions for the functional equation with continuous 

Solved the equality problem for two-variable Cauchy and quasi-arithmetic means

Extended previous results by weakening regularity assumptions

Abstract

In this paper, we are dealing with the solution of the functional equation $$ \varphi\Big(\frac{x+y}2\Big)(f(x)-f(y))=F(x)-F(y), $$ concerning the unknown functions $\varphi,f$ and $F$ defined on a same open subinterval of the reals. Improving the previous results related to this topic, we describe the solution triplets $(\varphi,f,F)$ assuming only the continuity of $\varphi$. As an application, under natural conditions, we also solve the equality problem of two-variable Cauchy means and two-variable quasi-arithmetic means.