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Homework Help: Complex analysis - Cauchy estimates

  1. Jun 6, 2008 #1
    1. The problem statement, all variables and given/known data
    Let [tex]D\subset\mathbb{C}[/tex] be the unitdisc and [tex]F=\{f:D\rightarrow D\,|\,\forall z\in D\partial_{\bar{z}}f=0\}[/tex], calculate [tex]L=\sup_{f\in F}|f''(0)|[/tex]. Show that there is an [tex]g\in F[/tex] with [tex]g''(0)=L[/tex].
    I am a bit stuck. But I think that it might be an idea to start with Cauchy estimate. Any other ideas?

    2. Relevant equations

    3. The attempt at a solution
    [tex]
    |a_2|\leq\frac{1}{2\pi}\frac{M}{r^3}l=\frac{M}{r^2}\quad M=\max_{|z|<r<1}|f(z)|=\sup_{z\in\partial D_r}|f(z)|=1?
    [/tex]
     
  2. jcsd
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