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In this paper it is shown that a new improvement on Hilbert's integral inequality can be established by introducing a weight function of the form 1 1+ √ x − 1 1+x (with x 0) .A s

It is shown that a weighted Hilbert inequality for double series can be established by introducing a proper weight function. Thus, a quite sharp result of the classical Hilbert inequality for double series is obtained. And a similar result for the Hilbert integral inequality is also proved. Some applications are considered.

It is shown that some new extensions of Hilbert's integral inequality with parameterλ(λ>1/2)can be established by introducing a proper weight function. In particular, whenλ=1, a refinement of Hilbert's integral inequality is obtained. As applications, some new extensions of Widder's inequality and Hardy-Littlewood's inequality are given.

The main objective of this paper is a study of some new generalizations of Hilbert’s type inequalities. More precisely, we obtain, in some general cases, that the constants involved in the right-hand sides of such inequalities are the best possible. Mathematics subject classification (2000): 26D15.

In this paper, the weight coefficient of the form π − θ (n)/ 2n + 1 (with θ ( n ) > 0) is introduced. Improvements on Hilbert's inequality and the Hardy–Littlewood inequality are established, and these results are extended.

This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real functionu(x)and a parameterλ. In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality inHpfunction are given.