Korean J. Math. Vol. 29 No. 3 (2021) pp.631-638
DOI: https://doi.org/10.11568/kjm.2021.29.3.631

Inequalities concerning polynomial and its derivative

Article Sidebar

PDF
Published: Sep 30, 2021
Keywords:
inequalities, polar derivative, modulus, coefficients
Mathematics Subject Classification:
26D10, 30C15, 41A17

Main Article Content

Bashir Ahmad Zargar
Department of Mathematics, University of Kashmir, Srinagar-190006, India
Manzoor Hussain Gulzar
Department of Mathematics, University of Kashmir, Srinagar-190006, India
Tawheeda Akhter
Department of Mathematics, University of Kashmir, Srinagar-190006, India

Abstract

In this paper, some sharp inequalities for ordinary derivative P′(z) and polar derivative DαP(z)=nP(z)+(α−z)P′(z) are obtained by including some of the coefficients and modulus of each individual zero of a polynomial P(z) of degree n not vanishing in the region |z|>k, k≥1. Our results also improve the bounds of Tur\'an's and Aziz's inequalities.



Article Details

References

[1] A. Aziz, Inequalities for the derivative of a polynomial, Proc. Amer. Math. Soc. 89 (1983), 259–266. Google Scholar

[2] A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivatives, J. Approx. Theory 54 (1998), 306–313. Google Scholar

[3] A. Aziz and N. A. Rather, Some Zygmund type Lq inequalities for polynomials, J. Math. Anal. Appl. 289 (2004), 14–29. Google Scholar

[4] A. Aziz and N. A. Rather, Inequalities for the polar derivative of a polynomial with restricted zeros, Math. Bulk. 17 (2003), 15–28. Google Scholar

[5] S. N. Bernstein, Sur e’ordre de la meilleure approximation des functions continues par des polynomes de degre’ donne’. Mem. Acad. R. Belg. 12 (4) (1912), 1–103. Google Scholar

[6] K. K. Dewan, J. Kaur and A. Mir, Inequalities for the derivative of a polynomial J. Math. Anal. Appl., 269 (2002), 489–499. Google Scholar

[7] V.N. Dubinin, Applications of Schwarz lemma to inequalities for entire functions with constraints on zeros, J. Math. Sci. 143 (2007), 3069–3076. Google Scholar

[8] N. K. Govil and P. Kumar, On Bernstein type inequalities for polar derivative of a polynomial, Prog. Approx. Theory Appl. Complex Anal Springer Optim Appl. 117 (2017), 41–74. Google Scholar

[9] N. K. Govil, P. Kumar, On sharpening of an inequality of Tur an, Appl. Anal. Discrete Math. 13 (2019), 711-720. Google Scholar

[10] P. Kumar, On the inequalities concerning polynomials, Complex Anal. and Oper. Theory 14 (2020), 1–11. Google Scholar

[11] P.Turan, U ̈ber die Ableitung von polynomen, Compos. Math.7 (1939), 89–95. Google Scholar