# Analytic function mapping inside the unit disk

I am working on this problem which appears to be a preliminary exam question in UTexas.

Assume that $f$ is analytic outside the closed unit disk and $|f(z)| <1$ in this region. Prove that $|f'(2)| \leq \frac{1}{3}$

I can prove that $|f'(2)| < 1$ but the upper bound $1/3$ is very hard to get. Any help would be appreciated.

Let $g(z):=f(1/z)$ then $g$ is holomorphic in $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ and $g( \mathbb{D})\subseteq \mathbb{D}$.
Then we can apply the Schwarz–Pick theorem (a variant of Schwarz lemma): for all $z\in \mathbb{D}$, $$|g'(z)|\leq \frac{1-|g(z)|^2}{1-|z|^2}.$$ Since $g'(z)=f'(1/z)\cdot(-1/z^2)$, it follows that $$|f'(1/z)|\leq \frac{(1-|f(1/z)|^2)\cdot|z|^2}{1-|z|^2}.$$ Now for $z=1/2$, we finally obtain $$|f'(2)|\leq \frac{(1-|f(2)|^2)\cdot|1/2|^2}{1-|1/2|^2}=\frac{1-|f(2)|^2}{3}\leq \frac{1}{3}.$$