On the location of zeros of generalized derivative

Document Type : Research Paper

Authors

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

Abstract

Let $P(z) =\displaystyle \prod_{v=1}^n (z-z_v),$ be a monic polynomial of degree $n$, then, $G_\gamma[P(z)] = \displaystyle \sum_{k=1}^n \gamma_k \prod_{{v=1},{v \neq k}}^n (z-z_v),$ where $\gamma= (\gamma_1,\gamma_2,\dots,\gamma_n)$ is a n-tuple of positive real numbers with $\sum_{k=1}^n \gamma_k = n$, be its generalized derivative. The classical Gauss-Lucas Theorem on the location of critical points have been extended to the class of generalized derivative\cite{g}. In this paper, we extend the Specht Theorem and the results proved by A.Aziz \cite{1} on the location of critical points to the class of generalized derivative .

Keywords

[1] A. Aziz, On the zeros of a Polynomials and its derivative; Bull. Aust. Math. Soc. 31(4) (1985) 245–255.
[2] J.Brown and G.Xiang, Proof of the Sendov Conjecture for the polynomial of degree atmost eight, J. Math, Anal, Appl, 232 (1999) 272–292.
[3] Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press,2002.
[4] N.A. Rather, A. Iqbal and I. Dar, On the zeros of a class of generalized derivatives, Rendi. Circ. Math. Palermo II. Ser (2020). https://doi.org/10.1007/s12215-020-00552-z
Volume 13, Issue 1
March 2022
Pages 179-184
  • Receive Date: 03 February 2021
  • Revise Date: 11 August 2021
  • Accept Date: 21 August 2021