TITLE: Bijective holomorphic map with a fixed point
QUESTION [5 upvotes]: $f$ 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.

REPLY [6 votes]: It's not true.  Consider $f(z) = z + z^2$ on $D = \mathbb C$.  The unique fixed point is $0$, $f'(0) = 1$, but $f(-1-z) = f(z)$.