Abstract
The famous Eneström-Kakeya theorem establishes upper bound for the moduli of the zeros of a polynomial p(z) with positive monotonic coefficients. In this paper we consider the Polar derivative of p(z) with certain notable restrictions on its coefficients and prove some glamorous results which provide generalizations and refinements of some well known results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aziz, A., and B.A. Zargar. 1996. Some extensions of Enestrom-Kakeya Theorem. Glasnik Matematicki 31: 239–244.
Aziz, A., and Q.G. Mohammad. 1984. Zero-free regions for polynomials and some generalizations of Enestrom-Kakeya theorem. Canadian Mathematical Bulletin 27: 265–272.
Dewan, K.K., and M. Bidkham. 1993. On the Enestrom-Kakeya theorem. Journal of Mathematical Analysis and Applications 180: 29–36.
Govil, N.K., Q.I. Rahman, and G. Schmeisser. 1979. On the derivative of a polynomial, Illinois. Journal of Mathematics 23: 319–329.
Joyal, A., G. Labelle, and Q.I. Rahman. 1967. On the location of zeros of polynomials. Canadian Mathematical Bulletin 10: 55–63.
Milovanovic, G.V., D.S. Mitrinovic, and Th.M. Rassias. 1994. Topics in polynomials: extremal properties, inequalities, zeros. Singapore: World scientific Publishing Co.
Marden, M. 1966. Geometry of polynomials, 2nd ed. Providence: American Mathematical Society.
Ramulu, P., and G.L. Reddy. 2015. On the zeros of polar derivatives. International Journal of Recent Research in Mathematics Computer Science and Information Technology(IJRRMCSIT) 2 (1): 143–145.
Reddy, G.L., P. Ramulu, and C. Gangadhar. 2015. On the zeros of polar derivatives of polynomials. Journal of Research in Applied Mathematics (Quest Journals) 2 (4): 7–10.
Titchmarsh, E.C. 1939. The theory of functions, 2nd ed. London: Oxford University Press.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author declares that there is no conflict of interest.
Ethical approval
This article does not contain any material with human participants or animals performed by any of the authors.
Additional information
Communicated by Samy Ponnusamy.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gulzar, M.H., Zargar, B.A. & Malik, S.A. On the zeros of the polar derivative of a polynomial. J Anal 30, 101–117 (2022). https://doi.org/10.1007/s41478-021-00333-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41478-021-00333-6