I am interested in functions $f$ that for a given sequence of real positive numbers $\{a_i\}$ hold the equality $$\sum_i a_i f^{-1}\left(f(x_i)\right) = f^{-1}\left(\sum_i a_i f(x_i)\right)$$.
Clearly, $f(x)=x$ is one such example, but are there any others? What about the special case $a_i =1 \forall i$?