Let $f$ be a holomorphic function on $\Omega \subset \Bbb R^2 \equiv \Bbb C$ and $u$ be a subharmonic function on $f(\Omega)$. Then show that the function $u \circ f$ is harmonic on $\Omega$.
Since $u$ is subharmonic on $f(\Omega)$ it is quite clear that $u \circ f$ is subharmonic on $\Omega$ because the function $u \circ f$ satisfies sub mean value property on $\Omega$. If we can show that the function satisfies super mean value property on $\Omega$ then we are done with the proof. Because then the function satisfies mean value property on $\Omega$ and then by the converse of the mean value theorem we can say that $u \circ f$ is harmonic on $\Omega$.
But how can I show that $u \circ f$ satisfies super harmonic property on $\Omega$? I got stuck at this stage. Any Cooperation in solving this problem is appreciated.
Thank you very much.