If have got a $k$-fold connected surface $G$, which is bounded by n distinct, non-intersecting Jordan-curves. By Ahlfors it is known that there exists a unique function $\phi$ which maps G to the unit circle $\Delta$ and a previously picked point $p \in G$ is mapped to $0 \in \Delta$ with $\phi^{\prime}(p)$ is maximal.

I´m trying to show that the map $\phi$ is proper, by using that G is relatively compact.

Can someone give a hint how to show that?

If have got a $k$-fold connected surface $G$, which is bounded by n distinct, non-intersecting Jordan-curves. By Ahlfors it is known that there exists a unique function $\phi$ which maps G to the unit disc $\Delta$ and a previously picked point $p \in G$ is mapped to $0 \in \Delta$ with $\phi^{\prime}(p)$ is maximal.

I´m trying to show that the map $\phi$ is proper, by using that G is relatively compact.

Can someone give a hint how to show that?