On a functional-differential equation with quasi-arithmetic mean value

TL;DR

This paper characterizes all differentiable functions satisfying a specific functional-differential equation involving quasi-arithmetic means, expanding understanding of mean-related functional equations.

Contribution

It provides a complete description of solutions to a new class of functional-differential equations involving quasi-arithmetic means.

Findings

All solutions are explicitly characterized.

The solutions relate to specific functional forms of ta and ta' functions.

The results generalize previous mean value equations.

Abstract

In this paper we describe all differentiable functions $\varphi,\psi\colon E\to\mathbb{R}$ satisfying the functional-differential equation \begin{equation*} [\varphi(y) - \varphi(x)]\psi '\bigl(h(x,y)\bigr) = [\psi(y) - \psi(x)]\varphi '\bigl(h(x,y)\bigr), \end{equation*} for all $x,y\in E$, $x<y$, where $E \subseteq \mathbb{R}$ is a nonempty open interval, $h(\cdot,\cdot)$ is a quasi-arithmetic mean, i.e. $h(x,y)=H^{-1}(\alpha H (x)+\beta H (y))$, $x,y\in E$, for some differentiable and strictly monotone function $H\colon E \to H(E)$ and fixed $\alpha, \beta\in (0,1)$ with $\alpha+\beta=1$.