Conformal map between annulii Is there any conformal map between $D_1= \lbrace z \in \mathcal{C} \; ; \; 1 \leq |z| \leq 2 \rbrace$ and $D_2 = \lbrace z \in \mathcal{C} \; ; \; 1 \leq |z| \leq 3 \rbrace$. 
By the Schottky theorem there is not. But I am looking for some other  way. Some hints will be usefull.

One approach to this problem is to use the Schwarz reflection principle. Suppose $$f : \{1<|z|<3\}\to \{1<|z|<2\}$$ is   conformal. Replacing $f$ with $f(2/z)$ if necessary, we can assume that $f$ takes one inner boundary (unit circle) to the other inner boundary (also unit circle). The reflection across the inner circle extends $f$ to a conformal map $$\{3^{-1}<|z|<3\}\to \{2^{-1}<|z|<2\}$$ Extending over the inner circle again, we get a map $$\{3^{-3}<|z|<3\}\to \{2^{-3}<|z|<2\}$$ and so on. The result is a conformal map between punctured disks. The singularity at $0$ is removable, by boundedness. Being holomorphic, $f$ must be locally Lipschitz; it satisfies an inequality of the form $|f(z)|\le C|z|$ near $0$. But this contradicts it taking $3^{-n}$ to $2^{-n}$ for infinitely many integer values of $n$.
A   different approach is presented in  When can we find holomorphic bijections between annuli? 
