Maximum value of derivative of a holomorphic function

Question

Let $U$ be an open unit disk and $f: U \rightarrow U$ is holomorphic mapping, such that $f(1/2)=1/3$. How to find the maximum value of $|f'(1/2)|$?

I tried to use Schwarz lemma but I didn’t succeed.

Answer 1

$$|f'(a)|\leq \frac{1-|f(a)|^2}{1-|a|^2},\forall a\in U. $$

To prove this, Fix $z\in U$. Apply Schwarz's lemma to $\phi_{f(z)}\circ f\circ \phi_{z}^{-1}:U\to U$ which sends $0$ to $0$.

Here for $w\in U$ we have bijective holomorphic map $\phi_w:z\mapsto \frac{z-w}{1-\overline wz},\forall z\in U.$

Answer 2

Hint: Consider the function $g(z)=\frac{f-1/3}{1-f/3}\left(\frac{1/2-z}{1-z/2}\right)$.