= Solution
The domains are not <conformally equivalent>. Suppose that $F:\mathbb C\setminus\{0\}\to\{w:0<|w|<1\}$ were a conformal equivalence. Since $F$ is bounded near zero, the <Riemann removable singularity theorem> extends it holomorphically across zero. The extension is a bounded <entire function>, so the <Liouville theorem> makes it constant, contradicting bijectivity. This is the <punctured plane is not conformally equivalent to the punctured unit disc> obstruction.
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