Generalization of Certain Inequalities Concerning the Polar Derivative of a Polynomial

Download PDF


DOI: 10.46793/KgJMat2304.613H


In this paper, we prove some more general results concerning the maximum modulus of the polar derivative of a polynomial. A variety of interesting results follow as special cases from our results.


Polar derivative, maximum modulus, zeros, inequalities.


[1]   N. C. Ankeny and T. J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math. 5(2) (1955), 849–852.

[2]   A. Aziz and B. A. Zargar, Inequalities for a polynomial and its derivative, Math. Ineq. Appl. 4(1) (1998), 543–550.

[3]   S. Bernstein, Sur la limitation des derivées des polynomes, Comptes Rendus de l’ Academic des Sciences (Paris) 190 (1930), 338–340.

[4]   N. K. Govil, A. Liman and W. M. Shah, Some inequalities concerning derivative and maximum modulus of polynomials, Aust. J. Math. Anal. Appl. 8 (2011), 1–8.

[5]   N. K. Govil, M. A. Qazi and Q. I. Rahman, Inequalities describing the growth of polynomials not vanishing in a disk of prescribed radius, Math. Ineq. Appl. 6(3) (2003), 453–467.

[6]   V. K. Jain, Inequalities for a polynomial and its derivative, Proc. Acad. Sci. (Math. Sci.) 32(2) (1997), 45–52.

[7]   P. N. Kumar, On the generalization of polynomial inequalities in the complex domain, J. Contemp. Math. Anal. 50(1) (2015), 14–21.

[8]    A. Liman, R. N. Mohapatra and W. M. Shah, Inequalities for polynomials not vanishing in a disk, Appl. Math. Comput. 218(3) (2011), 949–955. https://DOI:10.1016/j.amc.2011.01.077

[9]   A. Liman, I. Q. Peer and W. M. Shah, On some inequalities concerning the polar derivative of a polynomial, Ramanujan J. 38(2) (2015), 349–360.

[10]   P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513. https://DOI:10.1090/S0002-9904-1944-08177-9

[11]   M. Marden, The TeXbook, Math. Surveys 3, Amer. Math. Soc., Providence, RI, 1966.

[12]   A. Mir, Some sharp upper bound estimates for the maximal modulus of polar derivative of a polynomial, Annali Dell’Universita’Di Ferrara 65 (2019), 327–336.

[13]   H. A. S. Mezerji, M. A. Baseri, M. Bidhkam and A. Zireh, Generalization of certain inequalities for a polynomial and its derivative, Lobachevskii J. Math. 33(1) (2012), 68–74.

[14]   Q. I. Rahman and G. Schmeisser, The TeXbook, Oxford University Press, New York, 2002.

[15]   M. Riesz, Über einen satz des Herrn Serge Bernstein, Acta Math. 40 (1916), 337–347.

[16]   J. Somsuwan and M. Nakprasit, Some bounds for the polar derivative of a polynomial, Int. J. Math. Math. Sci. (2018), Article ID 5034607.

[17]   W. M. Shah A generalization of a theorem of Paul Turán, J. Ramanujan Math. Soc. 1 (1996), 67–72.