Let be an analytic function mapping the unit disc into itself. Suppose that has two different fixed points in the unit disc , show that must be the identity function for all .
is called a fixed point for a function , if .
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.