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Thread: Uniform Convergence of the Sequence of Compositions

  1. #1
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    Uniform Convergence of the Sequence of Compositions

    Hey everyone,
    I need help with the following complex analysis problem. (It is from chapter 3 of Greene & Krantz):

    Let $\displaystyle \phi : D(0,1) \mapsto D(0,1)$ be given by $\displaystyle \phi(z) = z + a_2 z^2 + a_3 z^3 + ...$ (notice in particular that $\displaystyle \phi$ is holomorphic). Let $\displaystyle \phi_1(z) = \phi(z), \phi_2(z) = \phi(\phi(z))$ and so on. Suppose that the sequence $\displaystyle \{ \phi_j \}$ converges uniformly on compact sets. Show that $\displaystyle \phi(z) = z$.

    Any help is appreciated! Thank you
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  2. #2
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    Quote Originally Posted by DJDorianGray View Post
    Hey everyone,
    I need help with the following complex analysis problem. (It is from chapter 3 of Greene & Krantz):

    Let $\displaystyle \phi : D(0,1) \mapsto D(0,1)$ be given by $\displaystyle \phi(z) = z + a_2 z^2 + a_3 z^3 + ...$ (notice in particular that $\displaystyle \phi$ is holomorphic). Let $\displaystyle \phi_1(z) = \phi(z), \phi_2(z) = \phi(\phi(z))$ and so on. Suppose that the sequence $\displaystyle \{ \phi_j \}$ converges uniformly on compact sets. Show that $\displaystyle \phi(z) = z$.

    Any help is appreciated! Thank you
    I haven't tried it rigourosly, but here's a naive approach that might help: If $\displaystyle \phi (z) := \sum_{j=1}^{\infty } a_jz^j$ where $\displaystyle a_1=1$, substitute $\displaystyle \varphi (z)= \lim_{n\rightarrow \infty } \phi _n(z)$ and we get $\displaystyle \phi _n(z)=\phi _{n-1}(z)+\sum_{j=2}^{\infty } a_j(\phi_{n-1}(z))^j$ and taking the limit (this is the step have to worry about) $\displaystyle \varphi (z)=\varphi (z) + \sum_{j=2}^{\infty } a_j\varphi (z)^j$ and thus $\displaystyle a_j=0$ for all $\displaystyle j\geq 2$ which gives $\displaystyle \phi (z)=z$.

    PS. Great book that one, lots and lots of exercises.
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    What is $\displaystyle D(0,1)$?
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