Let be the Riemann map of a simply connected domain onto the open unit disk, normalized by and . Show that if is any analytic function on such that for , then , with equality only when is a constant multiple of .

Remark. This shows that is the Ahlfors function of corresponding to .

There is a hint in the back of the book that says to apply the Schwarz lemma to . However, I still do not see how to prove this. I need help with this. Thank you.