Let $\Omega$ be a simply connected domain whose boundary has more than one point and $z_0, z_1 \in \Omega$. $\mathscr F=\{f(z)|f:\Omega \rightarrow \mathbb D$, $f$ is holomorphic, $f(z_0)=0\}$ is a family of holomorphic functions.
Prove that there exists a surjective function $\phi\in\mathscr F$ s.t. $|\phi(z_1)|=sup|f(z_1)|$ ($f\in\mathscr F)$.
I guess the solution is similar to the proof of Riemann Mapping Theorem, but I have problem when I tried to prove $\phi$ is surjective. Assume that $c\in \mathbb D-\phi(\Omega)$, and then I should make a contradiction by proving $\exists\psi\in\mathscr F$ s.t. $|\psi(z_1)| \gt|\phi(z_1)|$. I don't know how to construct $\psi$.