The following example rules out a substantial class of functions $f$. Suppose that $f$ is symmetric with $f(a) > f(0)$ for some $a \neq 0$. Define random variable $X$ as follows: ${\rm P}(X=0) = {\rm P}(X=a) = {\rm P}(X=-a) = 1/3$. Then,
$\mu(X)=m(X)=0$, and ${\rm P}[f(X) = f(0)] = 1/3$,  ${\rm P}[f(X) = f(a)]=2/3$. However,
$\mu(f(X)) = (1/3)f(0) + (2/3)f(a) < f(a) = m(f(X))$.