Do there exist a (non-trivial) globally Lipschitz continuous function $g:\mathbf{R}\to\mathbf{R}$ and a non-decreasing function $f:\mathbf{R}_+\to\mathbf{R}_+$ such that the identity \begin{equation} g(\alpha f(\alpha)s) = f(\alpha) g(s) \end{equation} holds for all $\alpha>0$ and $s\in\mathbf{R}$?
Remark: I am pretty sure that for $f$ polynomial this cannot hold (as then $g$ would have to be less than $1$-homogeneous, and thus not globally Lipschitz).