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December 2017 On some results of M.A. Malik concerning polynomials
Imtiaz Hussain, Abdullah Mir
Funct. Approx. Comment. Math. 57(2): 143-149 (December 2017). DOI: 10.7169/facm/1620

Abstract

If $P(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\le k, k\le 1,$ Rather, Gulzar and Ahangar [8] proved that for every $\alpha \in \mathbb{C}$ with $|\alpha|\ge k$ and $\gamma >0,$ \[ n(|\alpha|-k)\Bigg\{\int_0^{2\pi}\Big|\frac{P(e^{i\theta})}{D_{\alpha}P(e^{i\theta})}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}&\leq \Bigg\{\int_0^{2\pi}\Big|1+ke^{i\theta}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}. \] In this paper, we shall obtain a result which generalizes and sharpens the above inequality by obtaining a bound that depends upon the location of all the zeros of $P(z)$ rather than just on the location of the zero of largest modulus.

Citation

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Imtiaz Hussain. Abdullah Mir. "On some results of M.A. Malik concerning polynomials." Funct. Approx. Comment. Math. 57 (2) 143 - 149, December 2017. https://doi.org/10.7169/facm/1620

Information

Published: December 2017
First available in Project Euclid: 28 March 2017

zbMATH: 06864169
MathSciNet: MR3732893
Digital Object Identifier: 10.7169/facm/1620

Subjects:
Primary: 30A10
Secondary: 30C10, 30C15

Rights: Copyright © 2017 Adam Mickiewicz University

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Vol.57 • No. 2 • December 2017
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