Injectivity of Holomorphic function

**EinStone** · March 31st 2011, 02:53 PM

Let U be an open disk around the origin in $\mathbb{C}$ .
Suppose $f:U \rightarrow \mathbb{C}$ is holomorphic on $U$ , $f(0) = 0$ and $f'(0) = 1$ .

I want to show that there exists a neighborhood $V$ of $0$ , $V \subset U$ , so that $f$ is injective on $V$ .

Anybody can help?

**Drexel28** · March 31st 2011, 03:45 PM

Originally Posted by EinStone

Let U be an open disk around the origin in $\mathbb{C}$ .
Suppose $f:U \rightarrow \mathbb{C}$ is holomorphic on $U$ , $f(0) = 0$ and $f'(0) = 1$ .

I want to show that there exists a neighborhood $V$ of $0$ , $V \subset U$ , so that $f$ is injective on $V$ .

Anybody can help?

Isn't there a nice complex analysis analogue of the implicit function theorem which takes care of this quite nicely?

**EinStone** · March 31st 2011, 09:55 PM

Yes, it seems like a general statement for a holomorphic function to be locally invertible if it has non vanishing derivative. Something like an inverse function theorem, but I cant find it anywhere, if someone has a proof of this fact or a reference would be great.

**Drexel28** · March 31st 2011, 10:14 PM

Originally Posted by EinStone

Yes, it seems like a general statement for a holomorphic function to be locally invertible if it has non vanishing derivative. Something like an inverse function theorem, but I cant find it anywhere, if someone has a proof of this fact or a reference would be great.

There is a proof in this book, page 26.

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