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?**

2more comments