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Complex Analysis - Normal Families

  1. Jun 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]U[/itex] be a domain in [itex]\mathbb{C}[/itex] with [itex]z_0 \in U[/itex]. Let [itex]\mathcal{F}[/itex] be the family of analytic functions [itex]f[/itex] in [itex]U[/itex] such that [itex]f(z_0) = -1[/itex] and [itex]f(U) \cap \mathbb{Q}_{\geq 0} = \emptyset[/itex], where [itex]\mathbb{Q}_{\geq0}[/itex] denotes the set of non-negative rational numbers. Is [itex]\mathcal{F}[/itex] a normal family?


    2. Relevant equations

    Montel's Theorem: A family of analytic functions on a domain is normal if it is uniformly bounded on compact subsets of the domain.

    3. The attempt at a solution
    I'm not sure that this is a normal family; if it's not, how would I prove it? Would I have to produce an explicit sequence of functions from the family which has no subsequence which converges uniformly (on compact subsets)?
    If it is normal, I presumably need to show that it is uniformly bounded on compact subsets, which I don't know how to do either.

    Thanks.
     
  2. jcsd
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