Some inequalities for the growth of rational functions with prescribed poles

Rather, Nisar Ahmad; Wani, Mohmmad Shafi; Bhat, Aijaz Ahmad

doi:10.22075/ijnaa.2022.27564.3652

Some inequalities for the growth of rational functions with prescribed poles

Document Type : Research Paper

Authors

Nisar Ahmad Rather

Mohmmad Shafi Wani

Aijaz Ahmad Bhat

Department of Mathematics, University of Kashmir, Srinagar-190006, India

10.22075/ijnaa.2022.27564.3652

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 paper, we prove some results concerning the growth of rational functions with prescribed poles by involving some of the coefficients of polynomial $f(z)$. Our results not only improve the results of N. A. Rather et al. [8], but also give the extension of some recent results concerning the growth of polynomials by Kumar and Milovanovic [3] to the rational functions with prescribed poles and we obtain the analogous results for such rational functions with restricted zeros.

Keywords

Rational functions

polynomials

inequalities