Properties of holomorphic functions (demonstration)

Question

I don't know how to do this demonstration:

"If f is an holomorphic function, and M $\in \mathbb{R}^+$, such that for $z \in \mathbb{C}$, $|f(z)| \leq M(1+ |z|^n)$, then f is a $n$ or less degree polynomial"

Thanks for your attention!

Answer 1

Pick some $R>0$. Cauchy's estimate gives $|f^{(k)}(0)| \le k! { M(1+R^n) \over R^k }$.

If $k>n$, we can let $R \to \infty$ to get $f^{(k)}(0) = 0$ for $k > n$.

The result follows from the power series expansion of $f$.

Answer 2

Hints: $z^{-n} f(z)$ has a removable singularity at $\infty$. What kind of singularity does $f(z)$ have there? After subtracting ..., use Liouville's theorem.