Improved Regularity for a Composite Functional equation Stemming from the Theory of Means

Abstract In this paper we describe the solutions of the functional equation $$\begin{aligned} F\Big (\frac{x+y}{2}\Big )+f_1(x)+f_2(y)= G \big (g_1(x)+g_2(y)) \end{aligned}$$ F ( x + y 2 ) + f 1 ( x ) + f 2 ( y ) = G ( g 1 ( x ) + g 2 ( y ) ) defined on an open subinterval of $$ {\mathbb {R}} $$ R . Improving previous results we assume differentiability on each involved function, eliminate a former condition on $$ g'_1 $$ g 1 ′ and $$ g'_2 \, $$ g 2 ′ , moreover we determine a brand new family of solutions. We also present a particular member of this class as an example. In order to achieve this, we strengthen known results about certain auxiliary functional equations as well.