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Homework Help: Analytic mapping of unit disc onto itself with two fixed pts.

  1. Apr 8, 2012 #1
    1. The problem statement, all variables and given/known data

    let f(z) be a 1-1 analytic mapping of unit disc |z|<1 onto itself with two fixed points in |z|<1 Show that f(z)=z

    2. Relevant equations


    3. The attempt at a solution

    I'm thinking there has to be a theorem or something that I am missing for this.. But i'm not really sure where to get started..

    To start, let a and b be the two fixed points in |z|<1, then f(a)=a and f(b)=b and im sure there is a way to show that the only function that does this is f(z)=z not sure how though...
  2. jcsd
  3. Apr 8, 2012 #2


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    Science Advisor
    Homework Helper

    Try thinking about the Schwarz lemma.
  4. Apr 9, 2012 #3
    Got it, Thanks!!
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