Inequalities for the Polar Derivative of a Polynomial

Gulshan Singh, W. M. Shah, Yash Paul
DOI: 10.4236/apm.2011.12006                

Abstract

If is a polynomial of degree , having all its zeros in |z|≤K, K≥1 , then it was proved by Aziz and Rather [2] that for every real or complex number with |a| ≥K, . In this paper, we sharpen above result for the polynomials p(z) of degree n＞3

Keywords

Polynomial, Inequality, Polar Derivative

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Aziz and Q. M. Dawood, “Inequalities for a Polynomial and its Derivative,” Journal of Approximation Theory, Vol. 54, No. 3, 1998, pp. 306-313.
[2]	A. Aziz and N. A. Rather, “A Re-finement of a Theorem of Paul Turán Concerning Polynomi-als,” Journal of Mathematical Inequality Application, Vol. 1, No. 2, 1998, pp. 231-238.
[3]	N. G. de Bruijn, “Inequalities Concerning Polynomials in the Complex Domain,” Nederl. Akad. Wetench. Proc. Ser. A, Vol. 50, 1947, pp. 1265-1272; Indagationes Mathematicae, Vol. 9, 1947, pp. 591-598.
[4]	K. K. Dewan, N. Singh and A. Mir, “Growth of Polyno-mials not Vanishing inside a Circle,” International Journal of Mathematical Analysis, Vol. 1, No. 11, 2007, pp. 529-538.
[5]	N. K. Govil, “On the Derivative of a Polynomial,” Pro-ceedings of the American Mathematical Society, Vol. 41, 1973, pp. 543-546. doi:10.1090/S0002-9939-1973-0325932-8
[6]	P. D. Lax, “Proof of a Conjecture of P. Erd?s on the Derivative of a Polynomial,” American Mathematical Society, Vol. 50, No. 8, 1994, pp. 509-513.
[7]	M. A. Malik, “On the Derivative of a Polynomial,” Journal of the London Mathematical Society, Vol. 2, No. 1, 1969, pp. 57-60. doi:10.1112/jlms/s2-1.1.57
[8]	Polya and G. Szeg?, “Aus-gaben und Lehratze ous der Analysis,” Springer-Verlag, Berlin, 1995.
[9]	W. M. Shah, “A Generalization of a Theorem of Paul Turán,” Journal Ramanujan Mathematical Society, Vol. 11, 1996, pp. 67-72.
[10]	P. Turán, “über die Ableitung von Polynomen,” Compositio Mathematica, Vol. 7, 1939, pp. 89-95.