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Without using Reimann Mapping theorem , How to argue conformal maps does not exist ?

1) For $mathbb C/${0}$to mathbb D$

I know first set is not simply connected .But need to argue without using RMT.

I tried to Liovellies theorem but as function is not entire, I could not applied.
I also tried reverse direction I know that any non vanishing function is of form $e^{f(z)}$ for some holomorphic function f(z).
Please suggest some natural strategy to tackle such problem?

Any Help will be appreciated

Let $f : mathbb C setminus {0} to mathbb D$ holomorphic. Then $0$ is an isolated singularity of $f$ and $f$ is bounded. Hence, $0$ is a removable singularity of $f$. Therefore there is a entire function $g$ such that $f(z)=g(z)$ for all $z ne 0$. Then $g$ is bounded and by Liouville $g$ is constant. Thus $f$ is constant.

You can extend such a map to the whole $mathbb{C}$ by the classification of singularities. So you get a contradiction from Louville's Theorem.

If a conformal map existed then it would be invertible, under which function the image of $mathbb D$ would have to be simply connected.