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Some <i>L<sub>p</sub></i> Inequalities for <i>B</i>-Operators
Department of mathematics, University of Kashmir, Harzarbal, Sringar, India.
Department of mathematics, University of Kashmir, Harzarbal, Sringar, India.
- 1 Department of mathematics, University of Kashmir, Harzarbal, Sringar, India.
- 2 Department of mathematics, University of Kashmir, Harzarbal, Sringar, India.
Applied Mathematics·Volume 04 (2013)·Pages 155–166·Published 25 January 2013·DOI10.4236/am.2013.41026
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Abstract
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 B n -operator with parameters in the sense of Rahman [2], and . Unfortunately the proof of this re sult 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 0 ≦ p ﹤ 1 as well.
KeywordsL<sup>p</sup>-InequalitiesB<sub>n</sub>-OperatorsPolynomials
- 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
- 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
- G. Pólya an G. Szeg?, “Aufgaben und Lehrs?tze aus der Analysis,” Springer-Verlag, Berlin, 1925.
- Q. I. Rahman and G. Schmessier, “Analytic Theory of Polynomials,” Claredon Press, Oxford, 2002.
- 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
- G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, “Topics in Polynomials: Extremal Properties, Inequalities,” Zeros, World Scientific Publishing Co., Singapore City, 1994.
- A. Zugmund, “A Remark on Conjugate Series,” Proceedings London Mathematical Society, Vol. 34, No. 2, 1932, pp. 292-400.
- 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
- Q. I. Rahman and G. Schmessier, “Les Inequalitués de Markoff et de Bernstein,” Presses Univ. Montréal, Montréal, Quebec, 1983.
- 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.
- 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.
- 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
- 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
- N. G. Brijn, “Inequalities Concerning Polynomials in the Complex Domain,” Nederlandse Akademie Van Wetenschappen, Vol. 50, 1947, pp. 1265-1272.
- 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