On an Operator Preserving Inequalities between Polynomials


Let be the class of polynomials of degree n and a family of operators that map into itself. For , we investigate the dependence of on the maximum modulus of on for arbitrary real or complex numbers , with , and , and present certain sharp operator preserving inequalities between polynomials.

Share and Cite:

N. Rather, M. Shah and M. Mir, "On an Operator Preserving Inequalities between Polynomials," Applied Mathematics, Vol. 3 No. 6, 2012, pp. 557-563. doi: 10.4236/am.2012.36085.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, “Topics in Polynomials: Extremal Properties, Inequalities, Zeros,” World scientific Publishing Co., Singapore, 1994.
[2] Q. I. Rahman and G. Schmessier, “Analytic theory of polynomials,” Claredon Press, Oxford,2002.
[3] A. C. Schaffer, “Inequalities of A. Markoff and S. Bernstein for Polynomials and Related Functions,” Bulletin of the American Mathematical Society, Vol. 47, No. 2, 1941, pp. 565-579. doi:10.1090/S0002-9904-1941-07510-5
[4] M. Riesz, “Uber Einen Satz des Herrn Serge Bernstein,” Acta Mathematica, Vol. 40, 1916, pp. 337-347. doi:10.1007/BF02418550
[5] G. Pólya and G. Szeg?, “Aufgaben und Lehrs?tze aus der Analysis,” Springer-Verlag, Berlin, 1925.
[6] P. D. Lax, “Proof of a Conjecture of P. Erd?s on the Derivative of a Polynomial,” Bulletin of the American Mathematical Society, Vol. 50, No. 8, 1944, pp. 509-513. doi:10.1090/S0002-9904-1944-08177-9
[7] N. C. Ankeny and T. J. Rivlin, “On a theorm of S. Bernstein,” Pacific Journal of Mathematics, Vol. 5, 1955, pp. 849-852.
[8] A. Aziz and N. A. Rather, “On an Inequality of S. Bernstein and Gauss-Lucas Theorem,” Analytic and Geometriv Inequalities, Kluwer Academic Pub., Dordrecht, 1999, 29-35. doi:10.1007/978-94-011-4577-0_3
[9] Q. I. Rahman, “Functions of Exponential Type,” Transactions of the American Mathematical Society, Vol. 135, 1969, pp. 295-309. doi:10.1090/S0002-9947-1969-0232938-X
[10] E. B. Saff and T. Sheil-Small, “Coefficient and Integral Mean Estimates for Algebraic and Trigonometric Polynomials with Restricted Zeros,” Journal of the London Mathematical Society, Vol. 9, No. 2, 1975, pp. 16-22
[11] M. Marden, “Geometry of Polynomials,” Surveys in Mathematics, No. 3, 1949.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.