×

Inequalities for the polar derivative of a polynomial. (English) Zbl 1185.26022

Summary: Let \(p(z)\) be a polynomial of degree \(n\) and for any real or complex number \(\alpha \), and let \(D_{\alpha }p(z)=np(z)+(\alpha - z)p^{\prime}(z)\) denote the polar derivative of the polynomial \(p(z)\) with respect to \(\alpha \). In this paper, we obtain new results concerning the maximum modulus of a polar derivative of a polynomial with restricted zeros. Our results generalize as well as improve upon some well-known polynomial inequalities.

MSC:

26D05 Inequalities for trigonometric functions and polynomials
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[1] Lax PD: Proof of a conjecture of P. Erdös on the derivative of a polynomial.Bulletin of the American Mathematical Society 1944, 50: 509-513. 10.1090/S0002-9904-1944-08177-9 · Zbl 0061.01802 · doi:10.1090/S0002-9904-1944-08177-9
[2] Turán P: Über die Ableitung von polynomen.Compositio Mathematica 1939, 7: 89-95. · JFM 65.0324.01
[3] Malik MA: On the derivative of a polynomial.Journal of the London Mathematical Society 1969, 1: 57-60. 10.1112/jlms/s2-1.1.57 · Zbl 0179.37901 · doi:10.1112/jlms/s2-1.1.57
[4] Govil NK: On the derivative of a polynomial.Proceedings of the American Mathematical Society 1973, 41: 543-546. 10.1090/S0002-9939-1973-0325932-8 · Zbl 0279.30004 · doi:10.1090/S0002-9939-1973-0325932-8
[5] Gardner RB, Govil NK, Weems A: Some results concerning rate of growth of polynomials.East Journal on Approximations 2004,10(3):301-312. · Zbl 1113.30007
[6] Aziz A, Rather NA: A refinement of a theorem of Paul Turán concerning polynomials.Mathematical Inequalities & Applications 1998,1(2):231-238. · Zbl 0911.30002 · doi:10.7153/mia-01-21
[7] Dewan KK, Upadhye CM: Inequalities for the polar derivative of a polynomial.Journal of Inequalities in Pure and Applied Mathematics 2008,9(4, article 119):-9. · Zbl 1171.30002
[8] Dewan KK, Singh N, Mir A: Extensions of some polynomial inequalities to the polar derivative.Journal of Mathematical Analysis and Applications 2009,352(2):807-815. 10.1016/j.jmaa.2008.10.056 · Zbl 1160.26315 · doi:10.1016/j.jmaa.2008.10.056
[9] Aziz A, Dawood QM: Inequalities for a polynomial and its derivative.Journal of Approximation Theory 1988,54(3):306-313. 10.1016/0021-9045(88)90006-8 · Zbl 0663.41019 · doi:10.1016/0021-9045(88)90006-8
[10] Chanam B, Dewan KK: Inequalities for a polynomial and its derivative.Journal of Mathematical Analysis and Applications 2007,336(1):171-179. 10.1016/j.jmaa.2007.02.029 · Zbl 1124.30002 · doi:10.1016/j.jmaa.2007.02.029
[11] Aziz A: Inequalities for the derivative of a polynomial.Proceedings of the American Mathematical Society 1983,89(2):259-266. 10.1090/S0002-9939-1983-0712634-5 · Zbl 0537.30002 · doi:10.1090/S0002-9939-1983-0712634-5
[12] Dewan KK, Kaur J, Mir A: Inequalities for the derivative of a polynomial.Journal of Mathematical Analysis and Applications 2002,269(2):489-499. 10.1016/S0022-247X(02)00030-6 · Zbl 1010.30001 · doi:10.1016/S0022-247X(02)00030-6
[13] Dewan KK, Singh N, Mir A: Growth of polynomials not vanishing inside a circle.International Journal of Mathematical Analysis 2007,1(9-12):529-538. · Zbl 1129.30302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.