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Maximum modulus problem

  1. May 22, 2010 #1
    1. The problem statement, all variables and given/known data
    Find all entire functions f(z) with the property that |zf(z)|<=1 for all z in C

    2. Relevant equations
    The maximum modulus principle says that the only functions that are entire and bounded are constant functions.

    3. The attempt at a solution
    I know that if f(z) is entire, then zf(z) is also entire. Thus, if it's modulus is bounded on C, then it must be constant. Thus, zf(z)=c, so that f(z)=c/z where c is a constant. But then, f is not entire. Am I doing something wrong? Or is the only function that satisfies this property zero?
  2. jcsd
  3. May 22, 2010 #2


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    Science Advisor
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    That looks right to me. Only f(z)=0 works.
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