Let $U$ be a simple connected subset of $\mathbb{C} $ such that $U\neq\mathbb{C}$. Suppose $f:U \rightarrow \mathbb{C}$ a holomorphic function such that $f(a)=a$. I'd like to show that $|f'(a)|\leq 1$.
It looks a little bit like a Schwarz Lemma. I can (and proably have to) use Riemann's mapping theorem and take holomorphic function g, such that $g^{-1}:D(0,1) \rightarrow U, g(a)=0$ then I quess it is sufficient to take function $h(z)= \frac{f(z)-a}{z-a} $ and $h(a)= f^{'}(a)$ and look closely at a function $H:D(0,1) \rightarrow \mathbb{C}$ $H(z) = h(g^{-1}(z))$.But I can't get anythng out of it. Any hints are greatly appreciated.