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2012 Inequalities for the Polar Derivative of a Polynomial
Ahmad Zireh
Abstr. Appl. Anal. 2012: 1-13 (2012). DOI: 10.1155/2012/181934

Abstract

For a polynomial p ( z ) of degree n , we consider an operator D α which map a polynomial p ( z ) into D α p ( z ) : = ( α - z ) p ' ( z ) + n p ( z ) with respect to α . It was proved by Liman et al. (2010) that if p ( z ) has no zeros in | z | < 1, then for all α ,     β C with | α | 1 ,     | β | 1 and | z | = 1 , | z D α p ( z ) + n β ( ( | α | - 1 ) / 2 ) p ( z ) | ( n / 2 ) { [ | α + β ( ( | α | - 1 ) / 2 ) | + | z + β ( ( | α | - 1 ) / 2 ) | ] m a x | z | = 1 | p ( z ) | - [ | α + β ( ( | α | - 1 ) / 2 ) | - | z + β ( ( | α | - 1 ) / 2 ) | ] m i n | z | = 1 | p ( z ) | } . In this paper we extend the above inequality for the polynomials having no zeros in | z | < k , where k 1 . Our result generalizes certain well-known polynomial inequalities.

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Ahmad Zireh. "Inequalities for the Polar Derivative of a Polynomial." Abstr. Appl. Anal. 2012 1 - 13, 2012. https://doi.org/10.1155/2012/181934

Information

Published: 2012
First available in Project Euclid: 14 December 2012

zbMATH: 1242.26040
MathSciNet: MR2926908
Digital Object Identifier: 10.1155/2012/181934

Rights: Copyright © 2012 Hindawi

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