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

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 $\alpha+\beta=1$.