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...

Thank you