I was stuck on the following problem:
Let $0 < r_{0} < r_{1}$ and $0 < R_{0} < R_{1}$. Let $G$ be the annulus $\{z \in \mathbb{C} : r_{0} < |z| <r_{1}\}.$ Suppose $f$ is holomorphic on the interior of $G$ and continuous on $\partial{G}$. Further suppose that $f$ has no zeros in $G$ and $|f(z)|=R_{i}$ for $|z|=r_{i}$ (for $i=0,1$). Show that $f$ maps $G$ into the annulus $\{z \in \mathbb{C} : R_{0} < |z| <R_{1}\}$.
I really have no idea on how to start. I'd appreciate any hints or solutions even those that rely on more powerful results such as the uniformization theorem.