# Proving an entire function is a polynomial under certain conditions

#### Bingk

##### New member
Hello,
This was an exam question which I wasn't sure how to solve:

Suppose [TEX]f[/TEX] is entire and [TEX]|f(z)| \leq C(1+ |z|)^n[/TEX] for all [TEX]z \in \mathbb{C}[/TEX] and for some [TEX]n \in \mathbb{N}[/TEX].
Prove that [TEX]f[/TEX] is a polynomial of degree less than or equal to [TEX]n[/TEX].

I know that f can be expressed as a power series, but I'm not sure how to show that the upper limit of the sum has to be less than or equal to n.

Thanks!

#### PaulRS

##### Member
Hints:

• $f^{(n)}(z)$ is entire for all $n\in{\mathbb{N}}$.
• $\frac{1}{2\pi\cdot i}\cdot \oint_\Gamma \frac{f(z)}{(z-w)^{n+1}}dz = \frac{f^{(n)}(w)}{n!}$ where $\Gamma$ is, say, a circle centered at $w$ of radius $R$.
• What can you say, then, about $f^{(n)}(w)$ for some $n$ ? (Hint: try to find a uniform bound for $f^{(n)}(w)$ on the whole plane)