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Báo cáo: A New Proof of Inequality for Continuous Linear Functionals

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 179695, 9 pages doi:10.1155/2011/179695 Research Article A New Proof of Inequality for Continuous Linear Functionals Feng Cui and Shijun Yang College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China Correspondence should be addressed to Feng Cui, fcui@zjgsu.edu.cn Received 23 January 2011; Accepted 2 March 2011 Academic Editor: Andrei Volodin Copyright q 2011 F. Cui and S. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Gavrea and. | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 179695 9 pages doi 10.1155 2011 179695 Research Article A New Proof of Inequality for Continuous Linear Functionals Feng Cui and Shijun Yang College of Statistics and Mathematics Zhejiang Gongshang University Hangzhou 310018 China Correspondence should be addressed to Feng Cui fcui@zjgsu.edu.cn Received 23 January 2011 Accepted 2 March 2011 Academic Editor Andrei Volodin Copyright 2011 F. Cui and S. Yang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Gavrea and Ivan 2010 obtained an inequality for a continuous linear functional which annihilates all polynomials of degree at most k - 1 for some positive integer k. In this paper a new functional proof by Riesz representation theorem is provided. Related results and further applications of the inequality are also brought together. 1. Introduction Let k 1 be an integer and f e Ck a b . Denote by pk the set of all polynomials of degree not exceeding k. Let L C a b D be a continuous linear functional which annihilates all polynomials of degree at most k - 1 that is Lf 0 Vf ePk-1. 1.1 It is well known that a continuous linear functional is bounded and finding the bound or norm of a continuous linear functional is a fundamental task in functional analysis. Recently in light of the Taylor formula and the Cauchy-Schwarz inequality Gavrea and Ivan in 1 obtained an inequality for the continuous linear functional L satisfying 1.1 . In order to state their result we need some more symbols. Recall that the L2 norm of a square integrable function f on a b is defined by llf IIl2 a b - 1.2 2 Journal of Inequalities and Applications and denote by t max t 0 the truncated power function. The notation Lt fi fisf means that the functional L is applied to f considered as a function .