Share This Article:

Some Lp Inequalities for B-Operators

Abstract Full-Text HTML XML Download Download as PDF (Size:303KB) PP. 155-166
DOI: 10.4236/am.2013.41026    2,719 Downloads   4,190 Views  


If P(z) is a polynomial of degree at most n having all its zeros in , then it was recently claimed by Shah and Liman ([1], estimates for the family of $B$-operators, Operators and Matrices, (2011), 79-87) that for every R≧1, p 1, where B is a Bn-operator with parameters in the sense of Rahman [2], and . Unfortunately the proof of this result is not correct. In this paper, we present certain more general sharp Lp-inequalities for Bn-operators which not only provide a correct proof of the above inequality as a special case but also extend them for 0p1 as well.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

N. Rather and S. Ahangar, "Some Lp Inequalities for B-Operators," Applied Mathematics, Vol. 4 No. 1, 2013, pp. 155-166. doi: 10.4236/am.2013.41026.


[1] W. M. Shah and A. Liman, “Integral Estimates for the Family of B-Operators,” Operator and Matrices, Vol. 5, No. 1, 2011, pp. 79-87. doi:10.7153/oam-05-04
[2] Q. I. Rahman, “Functions of Exponential Type,” Transactions of the American Society, Vol. 135, 1969, pp. 295-309. doi:10.1090/S0002-9947-1969-0232938-X
[3] G. Pólya an G. Szeg?, “Aufgaben und Lehrs?tze aus der Analysis,” Springer-Verlag, Berlin, 1925.
[4] Q. I. Rahman and G. Schmessier, “Analytic Theory of Polynomials,” Claredon Press, Oxford, 2002.
[5] A. C. Schaffer, “Inequalities of A. Markoff and S. Bernstein for Polynomials and Related Functions,” Bulletin of the American Mathematical Society, Vol. 47, No. 8, 1941, pp. 565-579. doi:10.1090/S0002-9904-1941-07510-5
[6] G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, “Topics in Polynomials: Extremal Properties, Inequalities,” Zeros, World Scientific Publishing Co., Singapore City, 1994.
[7] A. Zugmund, “A Remark on Conjugate Series,” Proceedings London Mathematical Society, Vol. 34, No. 2, 1932, pp. 292-400.
[8] G. H. Hardy, “The Mean Value of the Modulus of an Analytic Function,” Proceedings London Mathematical Society, Vol. 14, 1915, pp. 269-277. doi:10.1112/plms/s2_14.1.269
[9] Q. I. Rahman and G. Schmessier, “Les Inequalitués de Markoff et de Bernstein,” Presses Univ. Montréal, Montréal, Quebec, 1983.
[10] M. Riesz, “Formula d’Interpolation pour la Dérivée d’un Polynome Trigonométrique,” Comptes Rendus de l’Académie des Sciences, Vol. 158, 1914, pp. 1152-1254.
[11] V. V. Arestov, “On Integral Inequalities for Trigonometric Polynimials and Their Derivatives,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, Vol. 45, No. 1, 1981, pp. 3-22.
[12] P. D. Lax, “Proof of a Conjecture of P. Erdos on the Derivative of a Polynomial,” Bulletin of the American Mathematical Society, Vol. 50, No. 5, 1944, pp. 509-513. doi:10.1090/S0002-9904-1944-08177-9
[13] N. C. Ankeny and T. J. Rivilin, “On a Theorm of S. Bernstein,” Pacific Journal of Mathematics, Vol. 5, 1955, pp. 849-852. doi:10.2140/pjm.1955.5.849
[14] N. G. Brijn, “Inequalities Concerning Polynomials in the Complex Domain,” Nederlandse Akademie Van Wetenschappen, Vol. 50, 1947, pp. 1265-1272.
[15] Q. I. Rahman and G. Schmessier, “Lp Inequalities for Polynomials,” Journal of Approximation Theory, Vol. 53, No. 1, 1988, pp. 26-32. doi:10.1016/0021-9045(88)90073-1
[16] R. P. Boas Jr. and Q. I. Rahman, “Lp Inequalities for Polynomials and Entire Functions,” Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, 1962, pp. 34-39. doi:10.1007/BF00253927
[17] K. K. Dewan and N. K. Govil, “An Inequality for Self-Inversive Polynomials,” Journal of Mathematical Analysis and Applications, Vol. 45, 1983, p. 490. doi:10.1016/0022-247X(83)90122-1
[18] A. Aziz, “A New Proof of a Theorem of De Bruijn,” Proceedings of the American Mathematical Society, Vol. 106, No. 2, 1989, pp. 345-350. doi:10.1090/S0002-9939-1989-0933511-6
[19] A. Aziz and N. A. Rather, “Some Compact Generalizations of Zygmund-Type Inequalities for Polynomials,” Nonlinear Studies, Vol. 6, No. 2, 1999, pp. 241-255.
[20] M. Marden, “Geometry of Polynomials,” Mathematical Surveys, No. 3, American Mathematical Society, Providence, 1966.

comments powered by Disqus

Copyright © 2018 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.