Improved regularity for a composite functional equation stemming from the theory of means

TL;DR

This paper advances the understanding of solutions to a specific composite functional equation related to means, by relaxing previous assumptions, introducing new solution families, and strengthening auxiliary results.

Contribution

It provides a broader class of solutions to the functional equation by removing previous restrictions and introduces new solution families with strengthened auxiliary results.

Findings

Eliminated previous conditions on derivatives g'_1 and g'_2.

Identified a new family of solutions for the functional equation.

Provided an explicit example of a new solution family.

Abstract

In this paper we describe the solutions of the functional equation \begin{equation*} F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G \big(g_1(x)+g_2(y)) \end{equation*} defined on an open subinterval of $ \mathbb{R} $. Improving previous results we assume differentiability on each involved function, eliminate a former condition on $ g'_1 $ and $ g'_2$, moreover we determine a brand new family of solutions. We also present a particular member of this class as an example. In order to achieve this, we strengthen known results about certain auxiliary functional equations as well.