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Liouville-type problem (Complex analysis)

  1. Dec 10, 2011 #1
    1. The problem statement, all variables and given/known data
    If f(z) is an entire function such that f(z)/z is bounded for |z|>R, then f''(z_0) = 0 for all z_0.


    2. Relevant equations
    Liouville's theorem
    Cauchy estimates: Suppose f is analytic for |z-z_0| ≤ ρ. If |f(z)|≤ M for |z-z_0| = ρ then the mth derivative of f at z_0 is bounded by a constant given in the book.

    3. The attempt at a solution
    I'm supposed to adapt the proof of Liouville's theorem I learned, which is by using the Cauchy estimates.
    I don't see where to get the second derivative of f. I'm pretty sure that z/f(z) is bounded for |z|<1/R, and I tried using the Cauchy estimates but couldn't get anything.
     
  2. jcsd
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