Polar Derivative Versions of Polynomial Inequalities

Barchand Chanam

doi:10.4236/apm.2015.512068

Abstract

Let be a polynomial of degree n and for a complex number , let denote the polar derivative of the polynomial with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir [Inter. Jour. Math. and Math. Sci., 16 (2005), 2641-2645] to polar derivative. Besides, another result due to Dewan et al. [J. Math. Anal. Appl. 269 (2002), 489-499] is also extended to polar derivative.

Highlights

Introduction and Statements of the ResultsLet p ( z ) be a polynomial of degree n and denote by M ( p, r ) = max p ( z)
Inequality (1.1) can be sharpened if we restrict ourselves to the class of polynomials having no zeros in z < 1
It was conjectured by Erdösand later verified by Lax [2] that if p ( z ) ≠ 0 in z < 1, max p′( z) ≤ n max p ( z)

Summary

Introduction and Statements of the Results

Equality holds in (1.1) if and only if p ( z ) has all its zeros at the origin. Inequality (1.1) can be sharpened if we restrict ourselves to the class of polynomials having no zeros in z < 1. It was conjectured by Erdösand later verified by Lax [2] that if p ( z ) ≠ 0 in z < 1 , max p′( z) ≤ n max p ( z). Inequality (1.2) is the best possible and equality attains for p ( z )= α + β zn , α = β. (2015) Polar Derivative Versions of Polynomial Inequalities.

Chanam

Lemmas

Proof of the Theorems n