In fact it is about holomorphic functions:
Does every holomorphic function $f:\mathbb{D}\longrightarrow \mathbb{D}$ have a fixed point?Here $\mathbb{D}$ is the open unit disk centered around $0$ on the complex plane$\mathbb{C}$.
It is an exercise from Stein's book,COMPLEX ANALYSIS,exercise 12 to Chapter 8.
My idea is as follows:first extend $f$ to the closure of the unit disk $\bar{\mathbb{D}}$,i.e.$\tilde{f}:\bar{\mathbb{D}}\longrightarrow\bar{\mathbb{D}}$.$\bar{f}$is continuous.Then apply the method of proving the classical fixed point theory on the unit disk.Yet this way we can at most get that $\bar{f}$ has fixed points,not $f$.
Are there any flaws in the above sketch of proof?
Or would someone be kind enough to give some hints on this problem?Thank you very much!