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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, 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!

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    Considering GEdgar's counterexample,maybe this problem should revised to show that every holomorphic function $f:\bar{\mathbb{D}}\longrightarrow\bar{\mathbb{D}}$ has a fixed point,which becomes the classical fixed point problem.2011-09-24

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