Suppose $\Omega$ is a bounded domain in the plane whose boundary consist of $m+1$ disjoint analytic simple closed curves.
Let $f$ be holomorphic and nonconstant on a neighborhood of the closure of $\Omega$ such that
$$|f(z)|=1$$ for all $z$ in the boundary of $\Omega$.
If $m=0$, then the maximum principle applied to $f$ and $1/f$ implies that $f$ has at least one zero in $\Omega$.
What about the general case? I believe that $f$ must have at least $m+1$ zeros in $\Omega$, but I'm not able to prove it...
Thank you