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

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.

• 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 $f$. – 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.
• 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? – Mel May 12 '14 at 7:07