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Showing a sequence/series of holomorphic functions is holomorphic

  1. Nov 16, 2012 #1
    1. The problem statement, all variables and given/known data
    1) Suppose a sequence of functions [itex]f_n \in H(\Omega)\cap C(\overline{\Omega})[/itex] converges uniformly on [itex]\partial \Omega[/itex]. Prove that the sequence [itex]f_n[/itex] converges uniformly on [itex]\overline{\Omega}[/itex] to a function [itex]f \in H(\Omega)\cap C(\overline{\Omega})[/itex].

    2)Suppose [itex]f \in H(\mathbb{D})[/itex] and [itex]f(0)=0[/itex]. Show that [itex]\sum_{n=1}^{\infty}f(z^n)[/itex] defines a holomorphic function on [itex]\mathbb{D}[/itex].


    2. Relevant equations



    3. The attempt at a solution

    1) I think I need to use the uniform convergence on the boundary to find a common M>0 such that each [itex]|f_n| < M[/itex] by Maximum Modulus, from which it follows that I have uniform convergence on the disk.

    2) So f is analytic and has no constant term. I'm not quite sure how that helps or what I should even think about applying to this problem.
     
  2. jcsd
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