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$f$ be a holomorphic on a bounded domain $D$ with fixed point $z_0$. Could any one give a hint how to show the following:

$f$ is bijective iff $|f'(z_0)|=1$.

Well, I was thinking like to compose $f,f^2,\dots,f^n$ and apply some how $f^n$ also has $z_0$ as fixed point. Thank you for help.

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    yes then, I hope, edited.2012-08-14

1 Answers 1

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It's not true. Consider $f(z) = z + z^2$ on $D = \mathbb C$. The unique fixed point is $0$, $f'(0) = 1$, but $f(-1-z) = f(z)$.

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    It's not stated in the question, but from considering the iterates of $f$, it seems that the intention is that $f \colon D \to D$ is a holomorphic self-map of the bounded domain $D$, and in that case, the conclusion holds. If $D \subset \mathbb{C}$ is a bounded domain, $f\colon D\to D$ holomorphic with a fixed point $z_0\in D$, then $f$ is bijective if and only if $\lvert f'(z_0)\rvert = 1$. (cc @BunuelianTrick)2014-04-29