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Math Help - Injectivity of Holomorphic function

  1. #1
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    Injectivity of Holomorphic function

    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?
    Last edited by EinStone; March 31st 2011 at 04:37 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by EinStone View Post
    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?
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  3. #3
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    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.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by EinStone View Post
    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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