Proving that $f(z)$ is polynomial

Question

Given $R>0 , M>0$ Let $f(z)$ be a entire function such as $|f(z)|\leq M|z|^m$ for evey z such as $|z|>R$. Show that $f(z)$ is a polynomial of degree m or lower.

Answer 1

Like Daniel Fischer pointed out, you can use the Cauchy estimate; since $f$ is entire, it converges to its Taylor series $$f(z_0)=\Sigma_{n=0}^{\infty}\frac {f^n(z_0)}{n!(z-z_0)^n}$$ at every $z_0$ in $\mathbb C$ . Use the Cauchy estimate on $|f^n(z_0)|$, together with your bound to show that $f^k(z_0)=0$ for all $n >k$ for some positive integer $k$. (you may need to translate by $z_0$ to be able to use $R$).