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A complex problem

  1. Jun 21, 2010 #1
    Hi all.

    The problem is "Prove that a function which is analytic in the whole plane and satisfies an inequality |f(z)| < |z|^n for some n and sufficiently large |z| reduces to a polynomial." I do not understand what I need to show that the function reduces to a polynomial.

    Any help will be appreciated.
  2. jcsd
  3. Jun 21, 2010 #2
    A more general theorem holds,that no entire function dominates another.. Check out :
    http://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis [Broken])
    Last edited by a moderator: May 4, 2017
  4. Jun 22, 2010 #3
    Write [tex]f(z) = \sum_{m=0}^{\infty} a_m\ z^{m}[/tex]

    (which you can do since it's entire).

    Show that |f(z)| < |z|^n implies for some natural number N,

    [tex]a_m = 0[/tex]

    for any m >= N.
    Last edited: Jun 22, 2010
  5. Dec 30, 2010 #4
    You can apply Cauchy Integral Formula to show that f^(n+m)(a) = 0 for any complex number a in C and m>=1.
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