Suppose that f is meromorphic in $\mathbb{C}$ and that there are $ K \in \mathbb{N}$ e positive numbers C,K such that $|f(z)|< |z|^{k}, \ |z|> R.$ Show that f is rational.
To make an argument that has a finite number of points, I was thinking of making a polynomial estimate for the f-module, conclude that it has a finite number of poles inside the closed and compact R-ray ball. Then multiply the meroforphic function by a polynomial function so it will be correct analytic ? and without poles. I think it has to extend to every complex plane there will be no poles and it will be limited there uses Liouvile's theorem ... I wonder if my thinking is correct.
After multiplying by a polynomial to get rid of the poles, you have an analytic function which is again bounded by some $|z|^m$ for $|z|>R$. Use Cauchy's formula on circles of large radius to bound the Maclaurin series coefficient of $z^n$ for $n > m$.