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Complex Analysis Problem

  1. Sep 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Find all entire functions [tex]f[/tex] such that

    [tex] |f(z)|\leq e^{\textrm{Re}(z)}\quad\forall z\in\mathbb{C}[/tex]

    2. Relevant equations

    3. The attempt at a solution

    I tried using Nachbin's theorem for functions of exponential type. I also tried using the Cauchy integral formula to see if I could gain more information about [tex]f[/tex] but I could not solve the problem.
  2. jcsd
  3. Sep 24, 2009 #2


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    I think you are looking at it in an overly complicated way. Can you think of a way to use |e^z|=e^(Re(z))?
  4. Sep 25, 2009 #3
    I tried applying schwarz lemma to [tex]|f(z)|\leq |e^z|[/tex] i.e.

    [tex]\left|\frac{f(z)}{e^z}\right|\leq 1[/tex]

    But this did not give me much information about [tex]f[/tex]. What other Theorems from Complex Analysis could I use to gain information about [tex]f[/tex]?
  5. Sep 25, 2009 #4


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    Liouville's theorem!
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