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Application of Schwarz Lemma-Complex Analysis

  1. May 8, 2012 #1
    Application of Schwarz Lemma--Complex Analysis

    Let f be holomorphic in D = {z: |z| < 1} and suppose that |f(z)| ≤ M for all z in D.

    If f(z[itex]_{k}[/itex]) = 0 for 1 ≤ k ≤ n, show that

    |f(z)| ≤ M [itex]\prod |z - z_{k}|/|1 - \bar{z_{k}}z|[/itex] for k = 1 to n and |z| < 1.


    I am just very confused about how to begin this problem. We know that f is bounded and holomorphic, and I feel like Schwarz's lemma is relevant here, and maybe a Mobius transformation. Any help in getting me started would be much appreciated. Thanks!
     
  2. jcsd
  3. Jun 19, 2012 #2
    Re: Application of Schwarz Lemma--Complex Analysis


    I think you're right, yet I feel that what you want is actually the Schwarz-Pick Lemma, which you can find in number 3 in [1]

    DonAntonio

    [1] http://en.wikipedia.org/wiki/Schwarz_lemma#Schwarz.E2.80.93Pick_theorem
     
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