# Finding the largest possible value of $|f(2)|$ and |f'(1)|?

Hi I was doing the following problem:

Let $f$ be analytic in the right half plane $\{z\in\mathbb{C}:\text{Re}(z)>0\}$ with $|f(z)|<1$ for all $z$. If $f(1)=0$,

(a) What is he largest value of $|f(2)|$?

(b) What is the largest possible value of $|f'(1)|$?

I got so far the following

The transformation $T(z)=\frac{z-1}{z+1}$ maps $P=\{z\in\mathbb{C}:\text{Re}(z)>0\}$ to unit disk and hence $f\circ T^{-1}$ maps the unit disk to itself and $$f\circ T^{-1}(0)=f(T^{-1}(0))=f(1)=0$$ Therefore Schwarz lemma can be applied and by Schwarz lemma we have $|f\circ T^{-1}(z)|\le|z|$

Then $$|f(2)|=|f\circ T^{-1}\circ T(2)|=\Bigl|\:f\circ T^{-1}\Big(\frac{1}{3}\Big)\Bigr|\le\frac{1}{3}$$ Hence the largest possible value will be $\frac{1}{3}$

But I couldn't do part (b). Anyone can help me with that. Any help would be highly appreciated. Thanks in advance.

Also by Schwarz $|(f\circ T^{-1})'(0)|\le 1$.
• Thank you very much for your hint. Just to make sure I got $|f'(1)|\le\frac{1}{2}$. Am I right?