Polar Derivative Versions of Polynomial Inequalities

Barchand Chanam

doi:10.4236/apm.2015.512068

Home > Journals > Physics & Mathematics > APM

APM> Vol.5 No.12, October 2015

Share This Article:

Polar Derivative Versions of Polynomial Inequalities ()

Abstract Full-Text HTML XML

Download as PDF (Size:346KB) PP. 745-755

DOI: 10.4236/apm.2015.512068 1,544 Downloads 1,934 Views

Author(s) Leave a comment

Barchand Chanam^*

Affiliation(s)

Department of Basic Sciences and Humanities, National Institute of Technology, Manipur, India.

ABSTRACT

Let

be a polynomial of degree n and for a complex number , let

denote the polar derivative of the polynomial

with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir [Inter. Jour. Math. and Math. Sci., 16 (2005), 2641-2645] to polar derivative. Besides, another result due to Dewan et al. [J. Math. Anal. Appl. 269 (2002), 489-499] is also extended to polar derivative.

KEYWORDS

Polynomials, Polar Derivative of a Polynomial, Zeros, Extremal Polynomials

Cite this paper

Chanam, B. (2015) Polar Derivative Versions of Polynomial Inequalities. Advances in Pure Mathematics, 5, 745-755. doi: 10.4236/apm.2015.512068.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Bernstein, S. (1926) Lecons Sur Les Propriétés extrémales et la meilleure approximation desfonctions analytiques d’une variable réele, Paris.

[2]	Lax, P.D. (1944) Proof of a Conjecture of P. Erdös on the Derivative of a Polynomial. Bulletin of the American Mathematical Society, 50, 509-513. http://dx.doi.org/10.1090/S0002-9904-1944-08177-9

[3]	Malik, M.A. (1969) On the Derivative of a Polynomial. Journal of the London Mathematical Society, 1, 57-60. http://dx.doi.org/10.1112/jlms/s2-1.1.57

[4]	Bidkham, M. and Dewan, K.K. (1992) Inequalities for a Polynomial and Its Derivative. Journal of Mathematical Analysis and Applications, 166, 319-324. http://dx.doi.org/10.1016/0022-247X(92)90298-R

[5]	Dewan, K.K. and Mir, A. (2005) On the Maximum Modulus of a Polynomial and Its Derivatives. International Journal of Mathematics and Mathematical Sciences, 16, 2641-2645. http://dx.doi.org/10.1155/IJMMS.2005.2641

[6]	Aziz (1983) Inequalities for the Derivatives of a Polynomial. Proceedings of the American Mathematical Society, 89, 259-266. http://dx.doi.org/10.1090/S0002-9939-1983-0712634-5

[7]	Turán, P. (1939) über die ableitung von polynomen. Compositio Mathematica, 7, 89-95.

[8]	Govil, N.K. (1973) On the Derivative of Polynomial. Proceedings of the American Mathematical Society, 41, 543-546. http://dx.doi.org/10.1090/S0002-9939-1973-0325932-8

[9]	Dewan, K.K., Kaur, J. and Mir, A. (2002) Inequalities for the Derivative of a Polynomial. Journal of Mathematical Analysis and Applications, 269, 489-499. http://dx.doi.org/10.1016/S0022-247X(02)00030-6

[10]	Aziz, A. and Rather, N.A. (1998) A Refinement of a Theorem of Paul Turán Concerning Polynomials. Mathematical Inequalities & Applications, 1, 231-238. http://dx.doi.org/10.7153/mia-01-21

[11]	Rather, N.A. (1998) Extremal Properties and Location of the Zeros of Polynomials. Ph.D. Thesis, University of Kashmir, Srinagar.

[12]	Govil, N.K., Rahman, Q.I. and Schmeisser, G. (1979) On the Derivative of a Polynomial. Illinois Journal of Mathematics, 23, 319-329.

[13]	Jain, V.K. (1994) Converse of an Extremal Problem in Polynomials II. Jourmal of the Indian Mathematical Society, 60, 41-47.

[14]	Qazi, M.A. (1992) On the Maximum Modulus of Polynomials. Proceedings of the American Mathematical Society, 115, 337-343. http://dx.doi.org/10.1090/S0002-9939-1992-1113648-1

[15]	Dewan, K.K. and Kaur, J. (1994) On the Maximum Modulus of Polynomials. Journal of Mathematical Analysis and Applications, 181, 493-497. http://dx.doi.org/10.1006/jmaa.1994.1038