Abstract and Applied Analysis

Inequalities for the Polar Derivative of a Polynomial

Ahmad Zireh

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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.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 181934, 13 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495737

Digital Object Identifier
doi:10.1155/2012/181934

Mathematical Reviews number (MathSciNet)
MR2926908

Zentralblatt MATH identifier
1242.26040

Citation

Zireh, Ahmad. Inequalities for the Polar Derivative of a Polynomial. Abstr. Appl. Anal. 2012 (2012), Article ID 181934, 13 pages. doi:10.1155/2012/181934. https://projecteuclid.org/euclid.aaa/1355495737


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