Functional equations involving means and their Gauss composition

In this paper the equivalence of the two functional equations

and

is studied, where $M_1$ and $M_2$ are two variable strict means on an open real interval $I$, and $M_1\otimes M_2$ denotes their Gauss composition. The equivalence of these equations is shown (without assuming further regularity assumptions on the unknown function $f:I\to{\mathbb R}$) for the cases when $M_1$ and $M_2$ are the arithmetic and geometric means, respectively, and also in the case when $M_1$, $M_2$, and $M_1\otimes M_2$ are quasi-arithmetic means. If $M_1$ and $M_2$ are weighted arithmetic means, then, depending on the algebraic character of the weight, the above equations can be equivalent and also non-equivalent to each other.