Riemann Mapping and Friends in a Vertical Strip

Question

This is an exercise about the Riemann mapping theorem, please give a hint on how to solve!

Let $S$ be $\{z:-1<\operatorname{Re}(z)<1\}$,

and $f:S \to S$ an analytic function satisfying $f(0)=0$.

Prove $|f'(0)| \leq 1$.

The claim looks like it comes from the Schwarz lemma. And regarding the Riemann mapping theorem, $f$ isn't said to be injective or surjective.

Answer 1

The Riemann mapping theorem can be used to prove the existence of a biholomorphic bijection from $S$ onto $D(0,1)$ such that $\psi(0)=0$. Let $F\colon D(0,1)\longrightarrow D(0,1)$ be equal to $\psi\circ f\circ\psi^{-1}$. Then you can apply the Schwarz lemma to prove that $\bigl|F'(0)\bigr|\leqslant1$ and it follows from this that $\bigl|f'(0)\bigr|\leqslant1$.