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On the location of zeros of generalized derivative | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 14، دوره 13، شماره 1، خرداد 2022، صفحه 179-184 اصل مقاله (319.3 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.22496.2382 | ||
| نویسندگان | ||
| Irfan Ahmad Wani* ؛ Mohammad Ibrahim Mir؛ Ishfaq Nazir | ||
| Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir, India | ||
| تاریخ دریافت: 15 بهمن 1399، تاریخ بازنگری: 20 مرداد 1400، تاریخ پذیرش: 30 مرداد 1400 | ||
| چکیده | ||
| 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 . | ||
| کلیدواژهها | ||
| polynomial؛ zeros؛ critical points and generalized derivative | ||
| مراجع | ||
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[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 at most 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 | ||
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