Composition of a subharmonic function and a conformal mapping

Question

this is q.4 of p.248 of Ahlfors book: Prove that a subharmonic function remains subharmonic after a composition with a conformal mapping. What I'v tried: Let $u:\Gamma \rightarrow \mathbb{R}$ and $C:\Gamma' \rightarrow \Gamma$ be the described maps, then we want to prove that if $u\circ C \leq h$ on $\partial M \subset\Gamma'$, $h$ harmonic, then $u\circ C \leq h$ on $M$, it is enough to prove that $C$ sends the boundary of $M$ to the boundary of $C(M)$ so then we can use the subharmonicity of $u$ to conclude, but I don't know if this is true.