On the zeros and critical points of a polynomial

Mathematical Analysis and its Contemporary Applications

دوره 4، شماره 1، فروردین 2022، صفحه 25-28   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.30495/maca.2021.1938758.1028

نویسندگان

Mohammad Ibrahim Mir؛ Irfan Ahmad Wani^* ؛ Ishfaq Nazir

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

چکیده

Let $P(z)=a_0 + a_1z + \dots  + a_{n-1}z^{n-1}+z^n$ be a polynomial of degree $n.$  The Gauss-Lucas Theorem asserts that the zeros of the derivative $P^\prime (z)= a_1 + \dots  +(n-1) a_{n-1}z^{n-2}+nz^{n-1},$  lie in the convex hull of the zeros of   $P(z).$ Given a zero of  $P(z)$ or $P^\prime (z),$  A. Aziz [1], determined regions which contain at least one zero of  $P(z)$ or $P^\prime (z)$ respectively. In this paper, we give simple proofs and improved version of various results proved in [1], concerning the zeros of a polynomial and its derivative.

کلیدواژه‌ها

polynomial؛ zeros؛ critical points؛ half plane؛ circular region