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

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