1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted