Abstract and Applied Analysis

Inequalities for the Polar Derivative of a Polynomial

Abstract

For a polynomial $p(z)$ of degree $n$, we consider an operator ${D}_{\alpha }$ which map a polynomial $p(z)$ into ${D}_{\alpha }p(z):=(\alpha -z)p\mathrm{\text{'}}(z)+np(z)$ with respect to $\alpha$. It was proved by Liman et al. (2010) that if $p(z)$ has no zeros in $|z|<1,$ then for all $\alpha ,\mathrm{ }\mathrm{ }\beta \in \mathbb{C}$ with $|\alpha |\ge 1, \mathrm{ }\mathrm{ }|\beta |\le 1$ and $|z|=1$, $|z{D}_{\alpha }p(z)+n\beta ((|\alpha |-1)/2)p(z)|\le (n/2)\{[|\alpha +\beta ((|\alpha |-1)/2)|+|z+\beta ((|\alpha |-1)/2)|]{\text{m}\text{a}\text{x}}_{|z|=1}|p(z)|-[|\alpha +\beta ((|\alpha |-1)/2)|-|z+\beta ((|\alpha |-1)/2)|]{\text{m}\text{i}\text{n}}_{|z|=1}|p(z)|\}$. In this paper we extend the above inequality for the polynomials having no zeros in $|z|, where $k\le 1$. 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