# Conformal mapping of region to unit disc

The complex text I'm reading asks the following question: find a conformal mapping of the region R between the circles,|z|=2 and |z-1|=1 onto the unit disc.

I know that the upper half plane can be mapped conformally into the unit disc by $$e^{i\theta}\frac{z-z_0}{z-\bar z_0}$$ where $$z_0$$ is in the upper half plane. So if I find a conformal mapping from R to the upper half plane, I can compose it with the previous map to get what I want. But I'm not sure how to proceed.

Hint: $$1/(z-2)$$ maps $$R$$ to a vertical strip, which can be mapped to the upper half plane.