On a functional equation related to two-variable weighted quasi-arithmetic means

In this paper, we are going to describe the solutions of the functional equation $$ \varphi\Big(\frac{x+y}{2}\Big)(f(x)+f(y))=\varphi(x)f(x)+\varphi(y)f(y) $$ concerning the unknown functions $\varphi$ and $f$ defined on an open interval. In our main result only the continuity of the function $\varphi$ and a regularity property of the set of zeroes of $f$ are assumed. As application, we determine the solutions of the functional equation $$ G(g(u)-g(v))=H(h(u)+h(v))+F(u)+F(v) $$ under monotonicity and differentiability conditions on the unknown functions $F,G,H,g,h$.