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They are the famous Hilbert integral inequalities, where the constant factors π2 and π are the best possible. And the equalities in (1.1) and (1.2) hold if and only if f (x) = 0, or g(x) = 0. These results can be found in papers [1] and [2]. Owing to the importance of the Hilbert inequality and the Hilbert type inequality in Mathematical analysis and applications, some mathematicians have been studying them. Recently, various refinements, extensions and generalizations of (1.2) appear in a great deal of the articles (such as [3]–[10] etc.). However, the research articles of (1.1) are few. The aim of the present paper is to build an inequality of the form

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 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.

A generalized Hardy-Hilbert inequality with weight function of the form B p−2+λ p , q−2+λ q �θr(λ)/(2n+1) λ − 2�λ r (with θr(λ) > 0,r = p, q ,1 � q 1) can be established by means of Euler-Maclaurin summation formula, where B(m,n) is β function. Inparticular, when λ = 1 ,animprovementonHardy-Hilbert'sinequality is obtained. As its applications, Hardy-Littlewood's inequality is extended and refined.

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, it is shown that the extended Hilbert’s inequality for double series can be refined by the aid of the Euler-Maclaurin summation formula. The extreme cases p → 1 + p\rightarrow 1^+ and q → + ∞ q\rightarrow +\infty are discussed.

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.

In this note, it is shown that the Hardy–Hilbert inequality for double series can be improved by introducing a proper weight function of the form π/sin(π/ p )− O r ( n )/ n 1−1/ r (with O r ( n )>0) into either of the two single summations. When r =2, the classical Hilbert inequality is improved.