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Proof somthing

  1. Jul 13, 2010 #1
    1. The problem statement, all variables and given/known data

    Hey guys.

    I've been sitting on this one for an hour or so. (got nothing)

    http://img138.imageshack.us/img138/765/24869754.png [Broken]

    I think it has something to do with schwarz lemma but I'm not sure.

    Any help will be much appreciate.

    Thanks a lot.

    2. Relevant equations

    3. The attempt at a solution
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jul 14, 2010 #2

    Gib Z

    User Avatar
    Homework Helper

    It is related to the Schwarz lemma, but its not directly applicable as we are not given f(0)=0. However, the function you have in an automorphism of the unit disk, and there's a way to "bring it back" to Schwarz's lemma.

    Theorem: Let [itex]f:D \to D[/itex] be an analytic automorphism of the unit disc and suppose [itex]f(\alpha) = 0[/itex]. Then there exists a real number [itex]\theta[/itex] such that:

    [tex]f(z) = \exp(i\theta) \frac{\alpha - z}{1-\overline{\alpha}z}[/tex]

    Start of proof:

    Let [tex]g=g_{\alpha}[/tex] be the above automorphism. Then [tex]h(w) = f(g^{-1}(w))[/tex] is an automorphism of the unit disc and maps 0 to 0. It suffices to prove h(w) has the form [tex]\exp(i\theta) w[/tex].

    To prove that, use the Schwarz Lemma in two different ways.
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