TITLE: On holomorphic branched coverings of a domain in the plane to the unit disk
QUESTION [6 upvotes]: This question is partly motivated by my answer to this question on math.stackexchange.
Let $\Omega$ be a bounded $n$-connected domain in the plane, bounded by $n$ pairwise disjoint Jordan curves. 
It was proved by Ahlfors that by solving an extremal problem (related to so-called analytic capacity), one can obtain a function $f$ holomorphic in $\Omega$ with the following properties :

$f$ is an $n$-to-$1$ branched covering of $\Omega$ onto the unit disk $\mathbb{D}$,
$f$ extends continuously to the boundary of $\Omega$, and maps each boundary curve homeomorphically onto the unit circle.

See e.g. Krantz's Geometric function theory: explorations in complex analysis, theorem 4.5.9.
My question is :
Is there another (perhaps more intuitive) way to see that such a function necessarily exists?

REPLY [2 votes]: I now know a lot more about this question than I did when I asked it, so for the sake of completness let me add :
By applying the Riemann mapping theorem $n$ times, we can assume that $\Omega$ is bounded by analytic Jordan curves. In this case, we have the following theorem due to Bieberbach, which says that there are many proper holomorphic maps onto the unit disk.
Theorem
Let $\Omega$ be a domain in $\mathbb{C}_{\infty}$ bounded by disjoint analytic Jordan curves $E_1, E_2, \dots, E_n$, and let $p \in X$. For each $j$, fix a point $\alpha_j \in E_j$. Then there is a unique proper holomorphic map $f:\Omega \to \mathbb{D}$ such that $f(\alpha_j)=1$ for all $j$ and $f(p)$=0.
For more information about proper holomorphic mappings, see the preprint there.