On the invariance of the arithmetic mean with respect to generalized Bajraktarević means

The purpose of this paper is to investigate the following invariance equation involving two $2$-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation $$f^{-1}\left(\frac{p_1(x)f(x)+p_2(y)f(y)}{p_1(x)+p_2(y)}\right)+g^{-1}\left(\frac{q_1(x)g(x)+q_2(y)g(y)}{q_1(x)+q_2(y)}\right)=x+y(x,y\in I),$$ where $I$ is a nonempty open real interval and $f,g:I\to \mathbb{R}$ are continuous, strictly monotone and $p_1,p_2,q_1,q_2:I\to {\mathbb{R}}_{+}$ are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of $f$, $g$, twice continuous differentiability of $p_1$ and $p_2$ and assuming that $p_1$ differs from $p_2$ on a dense subset of $I$, a necessary and sufficient condition for the equality above is that the unknown functions are of the form $$f=\frac{u}{v},g=\frac{w}{z},and{p}_1{q}_1=p_2{q}_2=vz,$$ where $u,v,w,z:I\to \mathbb{R}$ are arbitrary solutions of the second-order linear differential equation $F^{\mathrm{\prime }\mathrm{\prime }}=\gamma F$ ($\gamma \in \mathbb{R}$ is arbitrarily fixed) such that $v>0$ and $z>0$ holds on $I$ and $\{u,v\}$ and $\{w,z\}$ are linearly independent.