Generalization of Certain Inequalities Concerning the Polar Derivative of a Polynomial
![](../images/pdf.png)
Authors: I. HUSSAIN AND A. LIMAN
DOI: 10.46793/KgJMat2304.613H
Abstract:
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.
Keywords:
Polar derivative, maximum modulus, zeros, inequalities.
References:
[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. https://doi.org/10.3103/S1068362315010021
[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. https://doi.org/10.1007/s11139-014-9640-1
[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. https://doi.org/10.1007/s11565-019-00317-2
[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. https://doi.org/10.1007/BF02418550
[16] J. Somsuwan and M. Nakprasit, Some bounds for the polar derivative of a polynomial, Int. J. Math. Math. Sci. (2018), Article ID 5034607. https://doi.org/10.1155/2018/5034607
[17] W. M. Shah A generalization of a theorem of Paul Turán, J. Ramanujan Math. Soc. 1 (1996), 67–72.