Inequalities for rational functions with prescribed poles

TL;DR

This paper strengthens and refines classical polynomial inequalities for rational functions with prescribed poles, leading to improved bounds that consider polynomial coefficients and offering new insights into these inequalities.

Contribution

It introduces new techniques to extend polynomial inequalities to rational functions, providing refined bounds and applications to polynomial coefficient analysis.

Findings

Enhanced inequalities for rational functions with prescribed poles

Refined Erdős-Lax and Turán inequalities considering polynomial coefficients

New proofs of existing results without relying on previous lemmas

Abstract

For rational functions, we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erd\"os-Lax and Tur\'an to rational functions R. In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erd\'os-Lax and Tur\'an type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we shall demonstrate how some recently obtained results due to S. L. Wali and W. M. Shah, [Some applications of Dubinin's lemma to rational functions with prescribed poles, J. Math.Anal.Appl.450 (2017) 769-779], could have been proved without invoking the results