
conformal map, unit disk
Let $\displaystyle \phi(z)$ be a conformal map from a domain D onto the open unit disk $\displaystyle \mathbb{D}$. For $\displaystyle 0<r<1$, let $\displaystyle D_r$ be the set of $\displaystyle z \in D$ such that $\displaystyle \phi(z)<r$. Find a conformal map of $\displaystyle D_r$ onto $\displaystyle \mathbb{D}$.
The back of the book said that $\displaystyle \phi_r(z)=\frac{z}{r}$ works. I see how this map works. However, I don't see how to prove that this map is conformal. Thanks.

Being an analytic, univalent map it is certainly conformal!