Some Inequalities on Polar Derivative of Polynomial Having No Zero in a Disc

Barchand Chanam

doi:10.4236/apm.2015.513073

Abstract

Let , , be a polynomial of degree n having no zero in , , then Qazi [Proc. Amer. Math. Soc., 115 (1992), 337-343] proved . In this paper, we first extend the above inequality to polar derivative of a polynomial. Further, as an application of our result, we extend a result due to Dewan et al. [Southeast Asian Bull. Math., 27 (2003), 591-597] to polar derivative.

Highlights

Introduction and Statement of ResultsLet p ( z ) be a polynomial of degree n
If we restrict ourselves to the class of polynomials having no zero in z < 1, inequality (1.1) can be
It was conjectured by Erdös and later verified by Lax [2] that if p ( z ) ≠ 0 in z < 1, (1.1) can be replaced by max p′( z) ≤ n max p ( z)

Summary

Introduction

(2015) Some Inequalities on Polar Derivative of Polynomial Having No Zero in a Disc. Let p ( z ) be a polynomial of degree n and α be any real or complex number, the polar derivative of p ( z) , denoted by Dα p ( z ) , is defined as

Results

Conclusion