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.