Conformal map from $\mathbb C\setminus \{0\}$ to the open unit disk.

Question

Is there a conformal/analytic map from $\mathbb C\setminus \{0\}$ to the open unit disk?

I think the answer is no but I'm not sure how to prove it. I know that the former is not simply connected and the latter is simply-connected, but I'm not sure why this is useful or relevant since analytic maps are not necessarily homeomorphisms.

Answer 1

There's the constant map ...

Otherwise a holomorphic $f:\Bbb C-\{0\}\to D$ is bounded, so has a removable singularity at $0$, so is a restriction of a holomorphic $g:\Bbb C\to D$. Now apply Liouville.