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

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

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