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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.

In this article, it is shown that some new extensions on the Hardy-Hilbert inequality related to exponent function can be established by introducing a parameter λ (1 − q 1) and two exponent functions Aa x (A > 0,a > 1) and Bb y (B > 0,b > 1). In particular, for the case p = 2, an extension of the Hilbert integral inequality is built. As an application, a new Hardy-Littlewood integral inequality is 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.

The Cauchy–Schwarz inequality is improved by means of the positive definiteness of Gram's matrix. Heisenberg's inequality and Weyl's inequality are sharpened. A new inequality is established.

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.

A Hilbert-type integral inequality with parametersαand(α,λ>0)can be established by introducing a nonhomogeneous kernel function. And the constant factor is proved to be the best possible. And then some important and especial results are enumerated. As applications, some equivalent forms are studied.

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.