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

Abstract

In this paper we describe all differentiable functions $\phi ,\psi :E\to \mathbb{R}$ 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. $h(x,y)=H^{-1}(\alpha H(x)+\beta H(y))$, $x,y\in E$, for some differentiable and strictly monotone function $H:E\to H(E)$ and fixed $\alpha ,\beta \in (0,1)$ with $\alpha +\beta =1$.