# Injectivity of Holomorphic function

• Mar 31st 2011, 02:53 PM
EinStone
Injectivity of Holomorphic function
Let U be an open disk around the origin in $\displaystyle \mathbb{C}$.
Suppose $\displaystyle f:U \rightarrow \mathbb{C}$ is holomorphic on $\displaystyle U$ ,$\displaystyle f(0) = 0$ and $\displaystyle f'(0) = 1$.

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

Anybody can help?
• Mar 31st 2011, 03:45 PM
Drexel28
Quote:

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

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

Anybody can help?

Isn't there a nice complex analysis analogue of the implicit function theorem which takes care of this quite nicely?
• Mar 31st 2011, 09:55 PM
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.
• Mar 31st 2011, 10:14 PM
Drexel28
Quote:

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.