Inequalities Concerning Rational Functions With Prescribed Poles

TL;DR

This paper develops new inequalities for rational functions with fixed poles and restricted zeros, generalizing classical polynomial inequalities and providing refined bounds relevant to approximation theory.

Contribution

It introduces generalized inequalities for rational functions with prescribed poles and zeros, extending classical polynomial bounds and enhancing understanding in approximation theory.

Findings

Generalized inequalities for rational functions with fixed poles.

Refined bounds for rational functions with restricted zeros.

Connections to classical polynomial inequalities.

Abstract

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results for rational functions with fixed poles and restricted zeros. The obtained results bring forth generalizations and refinements of some known inequalities for rational functions and in turn produce generalizations and refinements of some polynomial inequalities as well.