A conformal mapping question.

Question

I am not sure about the following question, please help, thank you.

Find a conformal mapping $f$ of $D=\{|z|<1\}$ onto the domain $H=\{|\textrm{Im}(w)|<\pi/2\}$ such that $f(0)=0,\ f'(0)=2$.

Hence prove that for every $g$ analytic on $D$ such that $g(D)\subset H$ we have $g(0)=0\Rightarrow |g'(0)|\le 2$?

I have found that the map should be $\log(i \frac{w+1}{-w+1})$, but how do I argue that $g(0)=0\Rightarrow |g'(0)|\le 2$ in general? Is it related to the Riemann Mapping theorem?

Answer 1

Hint: Let $g$ be such a map. Apply Schwarz's lemma to $g \circ f^{-1}$.