On the Zeros of the Polar Derivative of a Polynomial

Document Type : Original Article


Department of Mathematics, University of Kashmir, Srinagar 190006, Jammu & Kashmir, India


Let P(z) be a polynomial of degree n whose coefficients satisfy an ≥ an−1 ≥ ... ≥ a0 > 0.Then according to the Enstrom-Kakeya Theorem, all the zeros of P(z) lie in |z|≤ 1. Aziz and Mohammad have shown that under the same condition on coeffients the zeros of P(z) whose modulus is greater than or equal to n/(n+1) are simple. In this paper, we extend the above result to the polar derivative.


[1]A.Aziz, Q.G.Mohammad, On the zeros of certain class of polynomials and related analytic functions,
J.Math.Anal.Appl.75(1980), 495-502.

[2]A.Aziz, Q.G.Mohammad, Zero free regions for polynomials and some generalizations of Enestrom-Kakeya Theorem,
Canad.Math.Bull.27(3),1984, 265-272.

[3]S. D. Bairagi, Viny Kumar Jain, T. K. Mishra, L. Saha, On the location of the zeros of certain polynomials,
Publications De L'Institut Mathematique, Nouvelle serie, tome 99(113)(2016)287-294.

[4]M. Marden, Geometry of polynomials, Math.Surveys No.3, Amer.Math.Soc., Providence, Rhode Island, 1966.

[5]M. H. Gulzar and A. W. Manzoor, On the zeros of the polar derivative of polynomials, International Journal of
Current Research, 8(2), (2016)26669-26674.

[6]M. H. Gulzar and Rubia Akhter, On the location of the zeros of certain polynomials, International Journal of
Advanced Scientifi c and Technical Research 8(3) (2018) 54-61.

[7]G. V. Milovanovic, D. S. Mitrinovic, T. M. Rassias, Topics in polynomials, Extremal problems, Inequalities, Zeros,
World Scientifi c, Singapore-New Jersy-London-Hongkong, 1994.

[8]P.Ramulu and G. L. Reddy, On the zeros of polar derivatives, International Journal of Recent Research in Mathematics,
Computer Science and Information Technology, Vol.2, Issue 1 (April2015-September2015), 143-145.
[9]G. L. Reddy, P. Ramulu and C. Gangadhar, On the zeros of polar derivative of polynomials, Journal of Research
in Applied Mathematics, 2(4), (2015) 4-7.

[10]Q. I. Rahman, G. Schmeisser, Analytic Theory Of Polynomials, Oxford University Press, Oxford, 2002.