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Thread: Complex Analysis, Schwarz lemma

  1. #1
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    Complex Analysis, Schwarz lemma

    Please help me with this one:

    $\displaystyle D=\left \{ z:\left | z \right |<1 \right \} $

    Let $\displaystyle f$ and $\displaystyle g$ be an analytical functions$\displaystyle f, g \rightarrow D $

    and also $\displaystyle f(0)=g(0), f'(0)=g'(0)$

    Prove that $\displaystyle f(z)=g(z), \forall z\in D$.

    Thanks!
    Last edited by sinichko; Dec 15th 2010 at 12:11 AM.
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  2. #2
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    Quote Originally Posted by sinichko View Post
    Please help me with this one:

    $\displaystyle D=\left \{ z:\left | z \right |<1 \right \} $

    Let $\displaystyle f$ and $\displaystyle g$ be an analytical functions$\displaystyle f, g \rightarrow D $

    and also $\displaystyle f(0)=g(0), f'(0)=g'(0)$

    Prove that $\displaystyle f(z)=g(z), \forall z\in D$.

    Thanks!
    Not true, for example $\displaystyle f(z) = z^2,\ g(z) = z^3.$
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by sinichko View Post
    Please help me with this one:

    $\displaystyle D=\left \{ z:\left | z \right |<1 \right \} $

    Let $\displaystyle f$ and $\displaystyle g$ be an analytical functions$\displaystyle f, g \rightarrow D $

    and also $\displaystyle f(0)=g(0), f'(0)=g'(0)$

    Prove that $\displaystyle f(z)=g(z), \forall z\in D$.

    Thanks!
    That statement is false, choose:

    $\displaystyle f(z)=z^2,\;g(z)=z^3$

    Fernando Revilla

    Edited: Sorry, I didn't see Opalg's post.
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  4. #4
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    See if you have the statement right.

    Meanwhile, a good strategy for any Schwarz problem is to construct a function satisfying the hypotheses of the lemma. Then everything frequently falls nicely onto your lap.

    You need an analytic function $\displaystyle h \to D$ sending 0 to 0.

    Try $\displaystyle h(z) = \frac{1}{2}\left[f(z) - g(z)\right].$.
    (Question: why did I scale by one half?)

    This may not lead where you want (especially since at the moment the conclusion you seek is unclear), but constructing something like this is usually a good first step.

    Verify that $\displaystyle h$ satisfies the hypotheses of the lemma and apply it. See what happens.
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