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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.
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