Let $f: \Omega \rightarrow \Omega$ be holomorphic where $\Omega \subset \mathbb{C}$ is a bounded region containing 0. If $f(0)=0$ and $f'(0)=1$, does $f(z)=z$?

This is true if $\Omega$ is a disk centered at 0 but does it hold if $D$ is only bounded?