Suppose Ω 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 Ω such that
|f(z)|=1 for all z in the boundary of Ω.
If m=0, then the maximum principle applied to f and 1/f implies that f has at least one zero in Ω.
What about the general case? I believe that f must have at least m+1 zeros in Ω, but I'm not able to prove it...