I am trying to prove:

If $f$ is entire and $f^{(5)}$ is bounded in $\mathbb{C}$, then $f$ is a polynomial of degree at most 5.

I think that Liouville's Theorem could be applicable here, however I don't think I can necessarily say $f^{(5)}$ is entire, which would be required to use Liouville's Theorem.

  • $\begingroup$ If $f$ is holomorphic at $z_0$, then $f'$ is also holomorphic at $z_0$ since it has a powerseries representation around $z_0$ with the same radius of convergence as $f$. $\endgroup$ – Dimitris May 12 '14 at 7:01

An entire function is infinitely differentiable and its derivatives are entire as well. Therefore, your idea is good: just apply Liouville's theorem to the function $f^{(5)}$: you get that $f^{(5)}$ is a constant, hence $f$ is a polynomial of degree at most 5. Hope this helps.

  • $\begingroup$ Thank you for the help. I understand intuitively why $f^{(5)}$ being constant implies that $f$ is a polynomial of degree at most 5, but how would I best prove this? $\endgroup$ – Mel May 12 '14 at 7:07
  • 1
    $\begingroup$ You are welcome. By the way, an holomorphic function is equal to its power series. Does this help you? $\endgroup$ – Romeo May 12 '14 at 7:34
  • $\begingroup$ Yes, that makes sense. Thanks again. $\endgroup$ – Mel May 12 '14 at 17:19
  • $\begingroup$ You are welcome. $\endgroup$ – Romeo May 12 '14 at 18:40

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