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 $P(z) = z \prod_{j=1}^{n-1}(z-z_{j})$, with $|z_{j}|\geq 1$, $1 \leq j \leq n-1.$ Then according to famous Brown's Conjecture $p'(z) \neq 0$, for $|z| < \frac{1}{n}.$ 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
[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
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Volume 12, Issue 2
November 2021Pages 1151-1155
- Receive Date: 04 December 2020
- Revise Date: 30 December 2020
- Accept Date: 08 January 2021