Let $f: \mathbb{C} \to \mathbb{C}$ be an entire function and $f^{(n)}$ is bounded. Show that $f$ is a polynomial with degree of $n$.

My Idea: So in somehow I have to use Liouville's Theorem. So we have to prove that $f$ is bounded.

  • 2
    $\begingroup$ Apply Liouville to $f^{(n)}$, not to $f$ ... $\endgroup$ – Martin R Jul 13 '17 at 15:07
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    $\begingroup$ First show that $f^{(n)}(x)$ is a constant using Liouville. Then integrate up $n$ times. $\endgroup$ – Winther Jul 13 '17 at 15:07
  • $\begingroup$ Closely related: math.stackexchange.com/questions/791306/…. $\endgroup$ – Martin R Jul 13 '17 at 15:09

Because of Liouville we can write $f^{(n)} (z)= c_n $ is konstant. Then \begin{align*} f^{(n-1)} (z) = c_n z + c_{n-1} \Rightarrow f^{(n-2)} = \frac{1}{2} c_n z^{2} + c_{n-1} z + c_{n-2} \Rightarrow \dots \Rightarrow f^{(0)}(z) = \sum_{k=0}^{n}\frac{c_k z^{k}}{k!} \end{align*}


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