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?
I assume that by $\Delta$ you mean the unit disk, not the unit cicle, as is written in the question.
There are several ways to prove that the Ahlfors map is a proper holomophic map of degree $n$ of $G$ onto $\Delta$. One of them is to use a variational argument (variation of zeros and boundary values), which was Ahlfors' original approach. For this, I suggest you take a look at the classical paper Bounded anaytic functions by Ahlfors.
Another way is to use the properties of the so-called Garabedian function on $G$ :
Suppose that $p=\infty$. Let $f$ be the Ahlfors map on $G$ for the point $\infty$, and let $\sigma$ be a measure on $\partial G$ of norm $\|\sigma\|=f'(\infty)$ such that $\int g d\sigma = g'(\infty)$ for all $g$ belonging to the class $A(G)$ of functions continuous on $\overline{G}$ and holomorphic in $G$. The measure $\mu=dz/(2\pi i) - \sigma$ is orthogonal to $A(G)$, so by a theorem of F. and M. Riesz we get that $\sigma = \psi(z) dz$ for some function $\psi$ that belongs to the Hardy space $H^1(G)$. This function $\psi$ is called the Garabedian function. It is not difficult then to see that the measure $f(z)\psi(z)dz$ is positive on $\partial G$, and the properness of the map $f$ can be deduced from this.
For more details on this approach, see the book Analytic capacity and measure by Garnett, Chapter 1, Theorem 4.1.
