ON THE HILBERT INEQUALITY

In this paper it is shown that the Hilbert inequality for double series can be improved by introducing a weight function of the form p n n+1 "p n 1 p n+1 ln n " , where n 2 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.