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Complex analysis - Schwarz's Lemma

  1. Apr 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that M|[itex]\frac{f(z)-f(z_{0})}{M^2-\overline{f(z_{0})}f(z)}[/itex]|≤|[itex]\frac{z-z_{0}}{1-\overline{z_{0}}z}[/itex]|

    for all z, z[itex]_{0}[/itex] in {w:|w|<1|

    2. Relevant equations

    I think I will have to use Schwarz's Lemma: if f is analytic in disc |z|<1, f(0)=0, and |f(z)|≤1 for all z in the disc, then |f(z)|≤|z| , |z|<1

    3. The attempt at a solution

    This looks similar to Schwarz Pick Theorem (found online, don't completely understand..)

    but one of the steps they used was to fix z which would mean that f(z)=z so the equations would become M|[itex]\frac{z-f(z_{0})}{M^2-\overline{f(z_{0})}z}[/itex]|≤|[itex]\frac{f(z)-z_{0}}{1-\overline{z_{0}}f(z)}[/itex]| or M|[itex]\frac{z-f(z_{0})}{M^2-\overline{f(z_{0})}z}[/itex]|≤|[itex]\frac{z-z_{0}}{1-\overline{z_{0}}z}[/itex]| not sure how that would help though...
     
  2. jcsd
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