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Homework Help: Bounded entire functions

  1. Mar 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f:C->C be an entire function such that Imf(z) <= 0 for all z in C. Prove that f is constant.

    2. Relevant equations
    Cauchy-Riemann equations??

    3. The attempt at a solution
    I don't know why I haven't been able to get anywhere with this problem. I feel like I have to use the fact that Imf(z) is harmonic or satisfies the Cauchy-Riemann equations, or something like that. And then somehow show that f is bounded. From there I just apply Liouville's Theorem. But I just need a slight push in the right direction. I mean, if Imf(z) <= 0 for all z, what does that say about its derivatives? This is really frustrating.
  2. jcsd
  3. Mar 16, 2009 #2
    Hint: Consider the function 1/(f(z)-i). Is it bounded?
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