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Math Help - liouville's theorem?

  1. #1
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    liouville's theorem?

    Suppose f is an entire function that satisfies  |f(z)|\leq M(1+|z|^m) for some constant M and positive integer m. Show that f is a polynomial of at most degree m.

    How should I approach this? Do I try to bound  |\frac{f(z)} {1+z^m}| , and then apply Liouville's theorem? The expression isn't necessarily entire though.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Use Cauchy estimates to show that

    \displaystyle  \bigg|\frac{\partial^{m+k} }{\partial z^{m+k}}f(z)\bigg|_{z=0}=0

    This will show that the power series of f about zero has only finitely many terms and is therefore a polynomial.
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  3. #3
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    Does this work?

    For 0\leq |z|\leq R, f(z)\leq M(1+ R^m) .

    Then I use the Cauchy estimate:

    For k>0, |f^{(m+k)}(0)|\leq \frac{(m+k)!M(1+R^m)}{R^{m+k}}. Then I use the trick from Liouville's theorem and take R to infinity, so f^{(m+k)}(0)=0. From the power series at 0, f has at most degree m.
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Yep that is the idea!
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