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Math Help - Analytic Function in the Unit Disk

  1. #1
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    Analytic Function in the Unit Disk

    Dear Colleagues,

    Could you please help me in solving the following function:

    Let f be analytic function in the unit disk D=\{z \in C:|z|<1\} with |f(z)|<1, for any z \in D. If f(z_{1})=z_{1}, f(z_{2})=z_{2} where z_{1} and z_{2} are distinct, show that f(z)=z for any z \in D.


    Best Regards,
    Raed.
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  2. #2
    Senior Member
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    Re: Analytic Function in the Unit Disk

    Use mobius transform g that maps 0 to z1, then apply Schwarz lemma - Wikipedia, the free encyclopedia to the composed function g^{-1}(f(g))
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