Let $D$ be a bounded domain, and let $f(z)$ be an analytic function from $D$ to $D$.Show that if $z_{0}$ is fixed point for $f(z)$,then $|f'(z_{0})|\leq 1$
All the conditions above make me think about Schwartz Lemma to solve this problem.But I don't know how to construct a proper function satisfying all the conditions in Schwartz Lemma.
Assume $D$ is simply connected. Let $\phi: D \to \Bbb U$ be a conformal map with $\phi(z_0)=0$, guaranteed by Riemann Mapping Theorem.
Define $g = \phi \circ f \circ \phi^{-1}: \Bbb U \to \Bbb U$. Then $g(0)=0$, so $|g'(0)| \leq 1$ by the Schwarz Lemma.
But $(\phi^{-1})'(z_0) = 1/\phi'(f(z_0))$ and therefore by the chain rule $$g'(0) = \phi'(f(z_0))f'(z_0)(\phi^{-1})'(0) = f'(z_0)$$ so that $|f'(z_0)|=|g'(0)| \leq 1$
