Let $f$ be a holomorphic one-to-one mapping of the annulus $1 ≤ |z| ≤ 2$, taking the inner boundary onto itself and the outer boundary onto itself. Show that $f$ is a rotation.
I considered function $g(z)=\frac{f(z)}{z}$ and have shown that it attains maximum at the boundary. How to show that it is constant by max-modulus principle?