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


1 Answer 1



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 $3$.


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