# A holomorphic function $f:U\to H$ where $U$ is the complement of closed discrete set must be constant.

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.

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.

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