Let $ \mathbb{D} = \{ z : |z|<1 \} $ and $ f $ an holomorphic function on $ \mathbb{D} $ and continuous on $ \overline{\mathbb{D}} $ such that $ f(\overline{\mathbb{D}}) \subset \mathbb{D} $.
Prove the following:
- There exists single point $ z^* \in \mathbb{D} $ such that $ f(z^*)=z^* $ (obvious by Rouche theorem).
- Let $ f_1=f,...,f_{n+1}=f(f_n) $ show that $ f_n(z) \longrightarrow z^* $ uniformly.