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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

  • 0
    Maybe use strong induction on $m$.2010-12-22
  • 0
    Also is the maximum principle the same thing as the maximum modulus theorem?2010-12-22
  • 0
    @Trevor, yes, I meant the maximum modulus theorem. I don't see how to use strong induction here..2010-12-30

4 Answers 4