# Describe class of rational functions such as $\left|f(z) \right| \leq M(1 + |z|^\pi),\:\: z\in \mathbb{C}$

I'm trying to describe class of rational functions with the following property: $$\exists M = M(f) > 0$$ and in the complex plane there is an estimate $$\left|f(z) \right| \leq M(1 + |z|^\pi),\:\: z\in \mathbb{C}$$

What have i try to do?

I tried to estimate the derivatives of the order higher than $$\pi$$ using the Cauchy integral formula $$f^{m} = \frac{m}{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{(\zeta - z)^{m+1}}d\zeta, \: \: z\in \mathbb{C}$$ and then i need to expand a function $$f$$ in a power series but i don't know how

UPD: Now I know the answer. It's class of polynomials of degree at most k, but how to get an answer

Hint

If $$f = \frac{h}{g}$$ where $$h, g$$ are coprimes then $$\deg g = 0$$. If not, $$g$$ has a root. What is the limit of $$f$$ at such a root?

Therefore $$f$$ is a polynomial. The inequality $$\left|f(x) \right| \leq M(1 + |z|^\pi),\:\: z\in \mathbb{C}$$

implies that $$\deg f \le 3 < \pi$$. Conversely, any polynomial of degree at most equal to $$3$$ satisfies such an inequality.

Finally the class $$\mathcal C$$ of requested functions is the one of polynomial of degree at most equal to $$3$$.