Let $w = f(v)$ where $f$ is a strictly increasing continuous function $[0;+\infty)\to [0;1)$. Let $A(w_1,\dots,w_n)$ be some average. Then we want $f^{-1}(A(f(cv_1),\dots,f(cv_n))) = cf^{-1}(A(f(v_1),\dots,f(v_n)))$ for every positive constant $c$.
Is this possible for some $f$ and $A$? Which one?
The question originated from this real world economical problem.
In a "good" solution the value $f^{-1}(A(f(v_1),\dots,f(v_n)))$ of should not go infinite when one of the values $v_k$ tends to plus infinity.