## Riemann map

Let $\phi(z)$ be the Riemann map of a simply connected domain $D$ onto the open unit disk, normalized by $\phi(z_0)=0$ and $\phi'(z_0)>0$. Show that if $f(z)$ is any analytic function on $D$ such that $|f(z)| \leq 1$ for $z \in D$, then $|f'(z_0)| \leq \phi'(z_0)$, with equality only when $f(z)$ is a constant multiple of $\phi(z)$.

Remark. This shows that $\phi(z)$ is the Ahlfors function of $D$ corresponding to $z_0$.

There is a hint in the back of the book that says to apply the Schwarz lemma to $f \circ \phi^{-1}$. However, I still do not see how to prove this. I need help with this. Thank you.