International Journal of Nonlinear Analysis and Applications

On generalisation of Brown's conjecture

Document Type : Research Paper

Authors

Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir, India

Abstract

Let  P be the complex polynomial of the form P(z)=z∏j=1n−1(z−zj), with |zj|≥1, 1≤j≤n−1. Then according to famous Brown's Conjecture p′(z)≠0, for |z|<1n. This conjecture was proved by Aziz and Zarger [1]. In this paper, we present some interesting generalisations of this conjecture and the results of several  authors related to this conjecture.

Keywords

[1] A. Aziz and B.A. Zarger, On the critical points of a polynomial, Aust. Math. Soc. 57 (1998) 173–174.
[2] B. A. Zarger and M. Ahmad, On some generalisation of Brown’s conjecture, Int. J. Nonlinear Anal. Appl. 7(2) (2016) 345–349.
[3] J.E. Brown, On the Ilief-Sendov conjecture, Pacific J. Math. 135 (1988) 223–232.
[4] M. Ibrahim, N. Ishfaq and I.A. Wani On zero free regions for derivatives of a polynomial, Krajugevac J. Math. 47(3) (2020) 403–407.
[5] Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002.
[6] N.A. Rather and F. Ahmad, On the critical points of a polynomial, BIBECHANA, 9 (2013) 28–32.
[7] B. Sendov, Generalization of a conjecture in the geometry of polynomials, Serdica Math. J. 28 (2002) 283–304.
[8] B.A Zarger and A.W. Manzoor, On zero free regions for the derivative of a polynomial, BIBECHANA, 14 (2017) 48–5
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 2
November 2021
Pages 1151-1155
  • Receive Date: 04 December 2020
  • Revise Date: 30 December 2020
  • Accept Date: 08 January 2021