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2012 Inequalities for the Polar Derivative of a Polynomial
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 ${D}_{\alpha }$ 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.

## Citation

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  