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Application of Liouville's Theorem (Complex Analysis)

  1. Jan 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that if [tex]f[/tex] is a meromorphic function [tex]f:\mathbb{C}\rightarrow\mathbb{C}[/tex] with

    [tex]|f(z)|^5\leq |z|^6\quad\textrm{for all}\quad z\in\mathbb{C}[/tex]

    Then [tex]f(z)=0[/tex] for all [tex]z\in\mathbb{C}[/tex]

    2. Relevant equations
    Liouville's Theorem

    A bounded entire function is constant.

    3. The attempt at a solution
    I tried applying Liouville's theorem to the quotient [tex]f(z)^5/z^6[/tex] which is bounded by 1 but was unsuccessful in proving that f is constant.
     
  2. jcsd
  3. Jan 11, 2010 #2
    Note that the hypothesis implies that [tex]f(z)^5/z^6[/tex] is a bounded function but because of the [tex]z^6[/tex] in the denominator, you must prove that it's also entire.

    To prove this, note (again, by the hypothesis) that [tex]f(z)[/tex] must have a zero at the origin, so either [tex]f(z)[/tex] is identically zero, or [tex]f(z) = z^ng(z)[/tex]; but then

    [tex]f(z)^5/z^6=z^{5n}g(z)^5/z^6[/tex]

    And this implies that [tex] n > 1 [/tex]. After simplifying, what can you say about [tex]g(z)[/tex]?
     
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