holomorphic funktion is a polynomial

Question

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be holomorphic. If we have $|f(z)|\leq|z|^n$ for some $n\in\mathbb{N}$ and all $z\in\mathbb{C}$, then $f$ is a polynomial.

I tried to apply Liouville's theorem but it does not help.

Thanks for your help.

Answer 1

$|f(z)|\leq|z|^n$ implies $f(0)=0$.

Writing $f(z)=z^m g(z)$ with $g(0)\ne0$ implies $m \ge n$ and so $|z^{m-n}g(z)| \le 1$.

Now apply Liouville's theorem.

Answer 2

Even true if the the condition is $|f(z)|\le C|z|^n$ for $|z|\ge R>0$. Let be $P_n$ the $n$th degree Taylor polynomial of $f$ at zero and consider $f-P_n$.