p. 305 - 312 On the Hilbert Inequality Zhou Yu and Gao Mingzhe Received: August 3, 2007; Accepted: February 21, 2008 Abstract. In this paper it is shown that the Hilbert inequality for double series~can be improved by introducing a weight function of the form (Ö n)/(n + 1) (((Ö n) – 1)/((Ö n)+1) – (ln n)/π) ), where n Î N. A similar result for the Hilbert integral inequality is also given. As applications, some sharp results of Hardy-Littlewood's theorem and Widder's theorem are obtained. Keywords: Hilbert's inequality; weight function; double series; monotonic function; Hardy-Littlewood's theorem; Widder's theorem. AMS Subject classification: Primary: 26D15. PDF Compressed Postscript Version to read ISSN 0862-9544 (Printed edition) Faculty of Mathematics, Physics and Informatics Comenius University 842 48 Bratislava, Slovak Republic Telephone: + 421-2-60295111 Fax: + 421-2-65425882 e-Mail: amuc@fmph.uniba.sk Internet: www.iam.fmph.uniba.sk/amuc © 2008, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE |