Let $\displaystyle f(z)$ be an analytic function mapping the unit disc $\displaystyle \mathbb{D}$ into itself. Suppose that $\displaystyle f(z)$ has two different fixed points in the unit disc $\displaystyle \mathbb{D}$, show that $\displaystyle f$ must be the identity function $\displaystyle f(z)=z$ for all $\displaystyle z \in \mathbb{D}$.

$\displaystyle z_0$ is called a fixed point for a function $\displaystyle f$, if $\displaystyle f(z_0)=z_0$.

For this one, I was thinking to use Rouché's Theorem or Schwarz lemma. However, we are dealing with fixed points here. So, I don't see what to do right now. I need a few hints on this one. Thanks.