Inequalities For The Growth Of Rational Functions With Prescribed Poles

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 $1\le j\le n$. 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.

I request to withdraw this article because I will correct and replace some results that have been published already. I am requesting to withdraw my whole submission till I add another another article