Using Rouché's theorem to prove the degree of a polynomial over $\mathbb{C}$

Question

Let $f$ be analytic over $\mathbb{C}$ and for $z\in\mathbb{C}, |f(z)|\le 7|z|^5$.

Prove that $f$ is a polynomial with degree $\le 5$.

Can Rouché's theorem be applied here? can I say that $7z^5$ has $5$ zeros on the plane and thus $f$ has maximum $5$ zeros meaning it's a polynomial with degree $\le 5$?

Answer 1

Hint: For $n\geq6$ consider the $n$-th coefficient of $f$ expansion in $|z|<r$, that is $$|a_n|=\left|\dfrac{1}{2\pi i}\int_{|z|=r}\dfrac{f(z)}{z^{n+1}}dz\right|\to0$$