Given a holomorphic map $f: \Omega\to \Omega$, where $\Omega$ is a simply-connected domain in $\mathbb{C}$, is the number of fixed points at most $1$ if $f$ is not the identity map? How many could they be?
By the Riemann Mapping Theorem, I am able to reduce the problem to finding a fixed point of a holomorphic map from the unit disc to itself. How should I proceed?
Thanks.