Inequalities For The Growth Of Rational Functions With Prescribed Poles

TL;DR

This paper studies the growth of rational functions with fixed poles, refining previous bounds and extending polynomial growth estimates to rational functions with prescribed poles, under certain zero restrictions.

Contribution

It generalizes polynomial growth estimates to rational functions with fixed poles and refines existing inequalities for their growth behavior.

Findings

Provides new bounds for rational functions with prescribed poles.

Extends polynomial growth estimates to a broader class of rational functions.

Refines previous results by strengthening inequalities and conditions.

Abstract

Let $\mathcal R_{n}$ be the set of all rational functions of the type $r(z) = f(z)/w(z)$, where $f(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-\beta_j)$, $|\beta_j|>1$ for $1\leq j\leq n$. In this work, we investigate the growth behavior of rational functions with prescribed poles by utilizing certain coefficients of the polynomial $f(z)$. The results obtained here not only refine and strengthen the findings of Rather et al. \cite{NS}, but also generalize recent growth estimates for polynomials due to Dhankhar and Kumar \cite{KD} to the broader setting of rational functions with fixed poles. Additionally, we establish corresponding results for such rational functions under suitable restrictions on their zeros.