International Journal of Nonlinear Analysis and Applications

On the zeros of polynomials and their generalized derivative

Articles in Press

Document Type : Research Paper

Authors

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

Abstract

In this paper, we present findings on the placement of zeros of generalized derivative of polynomials, drawing parallels to those observed in the ordinary derivative of polynomials. Mathematicians have broadened the scope of the Gauss-Lucas Theorem, a classic principle that deals with zero location in polynomials and their derivatives. The new work expands it to cover convex linear combinations of incomplete polynomials.

Keywords

[1] A. Aziz, On the zeros of a polynomials and its derivative, Bull. Aust. Math. Soc. 31 (1985), no. 4, 245–255.
[2] A.L. Cauchy, Exercises de mathematique, Oeuvres 9 (1829), 122. 2000.
[3] J.L. Dıaz-Barrero and J.J. Egozcue, A generalization of the Gauss-Lucas theorem, Czech. Math. J. 58 (2008), no. 2, 481–486.
[4] M. Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, American Mathematical Society, Rhode Island, 1966.
[5] M.I. Mir, I.A. Wani, and I. Nazir, On the zeros and critical points of a polynomial, Math. Anal. Contemp. Appl. 4 (2022), no. 1, 25–28.
[6] J. Sz-Nagy, Verallgemeinerung der derivierten in der geometric der polynome, Acta Univ. Szeged. Sect. Sci. Math. 13 (1950), 169–178.
[7] Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press Inc., New York, 2002.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 26 August 2025
  • Receive Date: 17 October 2024
  • Accept Date: 27 November 2024