how to prove that f is the identity map?

Question

Let $f:D\to D$ be a holomorphic function where $D=\{z\in C :|z|<1\}$ is the unit disc. Prove that if f has at least two fixed points then f is the identity map.

My opinion: I think if f(0)=0 in this case, we certainly can use Schwarz lemma to prove it. However, in this case, f(0) is not necessarily 0. If so, how can I prove it?

Answer 1

Let $T$be an automorphism of $\mathbb{D}$. Can you find a condition on $T$, so that $T^{-1} \circ f \circ T$ has the required property of having a fixed point at $0$? Now suppose there is another fixed point. Can Schwarz lemma be applied? (I'll leave you to find the appropriate $T$).