It seems to me that if $f$ is strictly monotonic, $m \circ f=f \circ m$, but when it is nonlinear, $\mu \circ f \ne f \circ \mu$. Those would be like a sufficient condition, at least.

It seems to me that if $f$ is strictly monotonic, $m \circ f=f \circ m$, and the question reduces to $\mu\circ f>f\circ\mu$, which is covered by Jensen's inequality. So strict convexity and strict monotonicity together would be a sufficient condition.

It seems to me that if $f$ is strictly monotonous, $m \circ f=f \circ m$, but when it is nonlinear, $\mu \circ f \ne f \circ \mu$. Those would be like a sufficient condition, at least.

It seems to me that if $f$ is strictly monotonic, $m \circ f=f \circ m$, but when it is nonlinear, $\mu \circ f \ne f \circ \mu$. Those would be like a sufficient condition, at least.