Abstract
Let \(P(z)=\sum \limits _{\nu =0}^na_\nu z^\nu\) be a polynomial of degree n and \(D_\alpha P(z)=nP(z)+(\alpha -z)P'(z)\) denotes the polar derivative of P(z), where \(\alpha\) being real or complex number. In this paper, we extend some existing results on the zeros of polar derivative of a polynomial under a restricted real coefficient conditions. Also, our results generalise many well-known results in this direction.
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 Q.G. Mohammad. 1980. On the zeros of certain class of polynomials and related analytic functions. Journal of Mathematical Analysis and applications 75: 495–502.
Aziz, A., and Q.G. Mohammad. 1984. Zero free regions for polynomials and some generalizations of Eneström–Kakeya Theorem. Canadian Mathematical Bulletin 27: 265–272.
Aziz, A., and B.A. Zargar. 1996. Some extensions of Eneström–Kaeya Theorem. Glasnik Mathematicki 31: 239–244.
Aziz, A., and B.A. Zargar. 2012. Bounds for the zeros of a polynomial with restricted coefficients. Journal of Applied Mathematics 3: 30–33.
Dewan, K.K., and M. Bidkham. 1993. On the Eneström–Kakeya Theorem. Journal of Mathematical Analysis and Applications 180: 29–36.
Joyal, L., and Q.I. Rahman. 1967. On the location of zeros of polynomial. Canadian Mathematical Bulletin 10: 53–63.
Milovanović, G.V., D.S. Mitrinović, 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 derivative of polynomials. Journal of Research in Applied Mathematics 2 (4): 7–10.
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
Zargar, B.A., Gulzar, M.H. & Akhter, T. On the zeros of polar derivative of polynomials with restricted coefficients. J Anal 30, 385–397 (2022). https://doi.org/10.1007/s41478-021-00348-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41478-021-00348-z