We say $f:D_1\to D_2$ is conformally equivalent if and only if $f$ is holomorphic and bijective.
For simply connected domain $D$, $f:D\to \Delta$ is holomorphic, $f$ is continuous in $\bar D$, and $f:\partial \bar D \to \partial \bar\Delta $ is homeomorphic, where $\Delta$ is unit disc, prove $f:D\to \Delta $ is conformally equivalent.
Now it is my approach, I suppose $f$ is injective, we need to prove $f$ is surjective, if not, then there exists $U$ is the proper subset of $\Delta$, $f$ can be prolongated from $\bar D$ to $\bar U$ homeomorphicly. Since $\bar U$ is the proper subset of $\bar\Delta$, $f(\partial \bar D )=\partial \bar U$, which is contradict to $f:\partial \bar D \to \partial \bar\Delta $.
So how should I prove $f$ is injective? Thanks very much!