$\{z : 0 < |z| < 1\}$ and $\{z : r < |z| < R\}$ are not conformally equivalent.

Question

Prove that $D=\{z : 0 < |z| < 1\}$ and $A=\{z : r < |z| < R\}$ are not conformally equivalent if $r > 0$.

I am trying to apply the Riemann mapping theorem to show but am not able to complete the proof...

Please give outlines of the proof.

Answer 1

Hint: Say $f:D\to A$ is a conformal equivalence. Then $f$ has a removable singularity at the origin, because... Now if $f(0)\in A$ then the Open Mapping Theorem shows that $f$ is not one-to-one in $D$, because... So $f(0)$ must lie on the boundary of $A$. This is a contradiction because...