Some inequalities for maximum modulus of rational functions

Abdullah Mir

doi:10.17951/a.2019.73.1.33-39

Abstract

In this paper, we establish some inequalities for rational functions with prescribed poles and restricted zeros in the sup-norm on the unit circle in the complex plane. Generalizations and refinements of rational function inequalities of Govil, Li, Mohapatra and Rodriguez are obtained.

Highlights

Let Pn denote the class of all complex algebraic polynomials P (z) of degree n
Rn is the set of rational functions with poles a1, a2, . . . , an at most and with a finite limit at ∞
Definition 1.1. (i) For P ∈ Pn, the conjugate transpose P ∗ of P is defined as P ∗(z) = znP

Summary

Introduction

Let Pn denote the class of all complex algebraic polynomials P (z) of degree n. Rn is the set of rational functions with poles a1, a2, . (i) For P ∈ Pn, the conjugate transpose P ∗ of P is defined as P ∗(z) = znP In 1995, Li, Mohapatra and Rodriguez [7] proved some inequalities similar to (1.1) and (1.3) for rational functions with poles outside the unit circle.

Results

Conclusion