AM  Vol.4 No.1 , January 2013
Some Lp Inequalities for B-Operators
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 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 0≦p﹤1 as well.
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.