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Let $S$ be a closed discrete subset of $\mathbb{C}$, let $U:=\mathbb{C}-S$ and let $H$ be the upper half plane. Let $f:U\to H$ be a holomorphic function. Then $f$ must be constant.

I am thinking of using the maximum modulus principle in combination with the open mapping theorem, but I don't see how can I prove the statement.

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Let $g (z)=\frac {1-if(z)} {1+if(z)}$. You can easily check that this maps $U$ into the open unit disk. It is holomorphic on $U$. Since it is bounded it has a removable singularity at each point of $S$. Hence it extends to a bounded entire function. By Louiville's Theorem it is a consatnt. This implies that $f$ is also a constant.

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  • $\begingroup$ Yeah this seems to do the trick, thanx. $\endgroup$ – Lucas Smits Jun 25 '19 at 9:07

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