Abstract
Let P(z) be a polynomial of degreen which does not vanish in ¦z¦ <k, wherek > 0. Fork ≤ 1, it is known that
, provided ¦P’(z)¦ and ¦Q’(z)¦ become maximum at the same point on ¦z¦ = 1, where\(Q(z) = z^n \overline {P(1/\bar z)} \). In this paper we obtain certain refinements of this result. We also present a refinement of a generalization of the theorem of Tuŕan.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aziz A, Inequalities for the derivative of a polynomial,Proc. Am. Math. Soc. 89 (1983) 259–266
Aziz A and Mohammad Q G, Simple proof of a theorem of Erdös and Lax,Proc. Am. Math. Soc. 80(1980) 119–122
Gardner R B and Govil N K, Inequalities concerning theL p norm of a polynomial and its derivative,J. Math. Anal. Appl. 179 (1993) 208–213
Govil N K, On a theorem of S Bernstein,Proc. Natl. Acad. Sci. 50 (1980) 50–52
Govil N K, Some inequalities for derivatives of polynomials,J. Approx. Theory 66(1991) 29–35
Govil N K and Rahman Q I, Functions of exponential type not vanishing in a half plane and related polynomial,Trans. Am. Math. Soc. 137(1969) 501–517
Lax P D, Proof of a conjecture of P Erdös on the derivative of a polynomial,Bull. Am. Math. Soc. 50(1944)509–513
Malik M A, On the derivative of a polynomial,J. London Math. Soc. 1 (1969) 57–60
Schaeffer A C, Inequalities of A Markoff and S Bernstein for polynomials and related functions,Bull. Am. Math. Soc. 47 (1941) 565–579
Tuŕan P, Über die Ableitung von PolynomenCompos. Math. 7(1939), 89–95
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Aziz, A., Ahmad, N. Inequalities for the derivative of a polynomial. Proc. Indian Acad. Sci. (Math. Sci.) 107, 189–196 (1997). https://doi.org/10.1007/BF02837727
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02837727