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

Abstract

In this paper we describe all differentiable functions $φ,ψ\colon E\to\mathbb{R}$ satisfying the functional-differential equation \begin{equation*} [φ(y) - φ(x)]ψ'\bigl(h(x,y)\bigr) = [ψ(y) - ψ(x)]φ'\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. $h(x,y)=H^{-1}(αH (x)+βH (y))$, $x,y\in E$, for some differentiable and strictly monotone function $H\colon E \to H(E)$ and fixed $α, β\in (0,1)$ with $α+β=1$.