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Let $ f : D \rightarrow D $ be holomorphic with $ f(0) = \frac{1}{2}$ and $ f(\frac{1}{2}) = 0 $, where $ D = \{ z : |z|\leq 1 \} $. Please suggest which of the following can be correct ..

  1. $ |f'(0)|\leq \frac{3}{4}$.

  2. $ |f'(1/2) |\leq \frac{4}{3}$.

  3. $ |f'(0)|\leq \frac{3}{4}$ and $ |f'(1/2)|\leq \frac{4}{3}$.

  4. $ f(z)=z$, $ z\in D$

Please help.

  • 2
    Are you sure you told us the whole question? There isn't an additional assumption, like $f(D) \subseteq D$?2012-06-10
  • 0
    @RobertIsrael: Thanks for pointing out the mistake. I have edited now.2012-06-10

3 Answers 3