Hilbert's Inequality Research Articles

The Generalizations of Hilbert's Inequality

By introducing some parameters and estimating the weight functions, we establish a new Hilbert-type inequalities with best constant factors. The equivalent inequalities are also considered.

On a New Weighted Hilbert Inequality

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.

New extensions of Hilbert’s inequality with multiple parameters

By introducing some parameters, we give new extensions of Hilbert inequality with best constant factors.

On inequalities of Hilbert's type

By introducing the function 1/(min{x, y}), we establish several new inequalities similar to Hilbert's type inequality. Moreover, some further unification of Hardy-Hilbert's and Hardy-Hilbert's type integral inequality and its equivalent form with the best constant factor are proved, which contain the classic Hilbert's inequality as special case.

On Further Analogs of Hilbert's Inequality

By introducing the function|lnx−lny|/(x+y+|x−y|), we establish new inequalities similar to Hilbert's type inequality for integrals. As applications, we give its equivalent form as well.

Hilbert's Type Linear Operator and Some Extensions of Hilbert's Inequality

The norm of a Hilbert's type linear operator is given. As applications, a new generalizations of Hilbert integral inequality, and the result of series analogues are given correspondingly.

A Hilbert inequality and an Euler-Maclaurin summation formula

AbstractWe obtain a generalized discrete Hilbert and Hardy-Hilbert inequality with non-conjugate parameters by means of an Euler-Maclaurin summation formula. We derive some general results for homogeneous functions and compare our findings with existing results. We improve some earlier results and apply the results to some special homogeneous functions.

On a more accurate Hilbert's type inequality

By using the improved Euler-Maclaurin’s summation formula, a more accurate Hilbert’s type inequality is built. As applications, the equivalent form is considered. All the lemmas and the theorem provide some new estimates on this type of inequalities. Mathematics Subject Classification: 26D15

A Hardy–Hilbert's type inequality with gamma function and its applications

In this paper, it is shown that a Hardy–Hilbert integral type inequality related to gamma function can be established by introducing a proper weight function of the form (Γ (x))1−λ (Γ′(x))1−r (x≥2, r>1, 1−(q/p)<λ≤2, p≥q>1). And the double series analogue of the Hardy–Hilbert type inequality is also built. In particular, for case p=2, some new extensions of the classical Hilbert's inequality (including integral and discrete forms) are obtained. As applications, some extensions on Hardy–Litlewood's theorem are given.

On new generalizations of the Hilbert integral inequality

Some new generalizations of the Hilbert integral inequality by introducing real functions ϕ ( x ) and ψ ( x ) . The results of this paper reduce to those of the corresponding inequalities proved by Gao [Mingzhe Gao, On Hilbert's integral inequality, Math. Appl. 11 (3) (1998) 32–35]. Some applications are considered.

Extension of Hilbert's inequality

In this paper we make some further extensions of discrete Hilbert's inequality by using Euler–Maclaurin summation formula. We give the improvements of some previously obtained results and also compare our results with some previously known from the literature.

Hilbert's inequalities and their reverses

In this paper, we make some further generalizations of well known Hilbert's inequality and its equivalent form in both integral and discrete case. A reverses of Hilbert's inequality are also given in integral case. Several other results of this type in recent years, follows as a special case of our results.

Some remarks on Hilbert's integral inequality

A generalization of the well-known Hilbert's inequality is given. Several other results of this type obtained in recent years follow as a special case of our result.

On a new extension of Hilbert's inequality

This paper deals with a new extension of Hilbert’s inequality with a (p, q)− parameter and a best constant factor. We also consider a more extended form and the equivalent inequality. Mathematics subject classification (2000): 26D15.

On the best constant in Hilbert's inequality

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 note on the weighted Hilbert’s inequality

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function which is a generalization of the classical Hilbert transformation. The weighted Hilbert inequality, which has applications in analytic number theory, is closely related to the finite Hilbert transformation associated with a polynomial. In this note, we study a relation between the finite Hilbert transformation and the weighted Hilbert’s inequality. A question is posed about the finite Hilbert transformation, of which an affirmative answer implies the weighted Hilbert inequality.

On new extensions of Hilbert's inequality

By introducing three parameters A, B and α, and estimating the weight coefficient, we give a new extension of Hilbert's inequality with a best constant factor involving the β function. As applications, we consider the equivalent form and some particular results.

An explicit formula for finite Hilbert transforms associated with a polynomial

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function, which is a generalization of the classical Hilbert transformation (cf. [4]). Hilbert's inequality, which has applications in analytic number theory (see Vaaler [11]), is closely related to the finite Hilbert transformation associated with a polynomial. In this paper, an explicit construction is given for the finite Hilbert transformation.

Generalization of Hilbert's Integral Inequality

Generalization of Hilbert's Integral Inequality

On the symmetric Hilbert's inequality and its applications

In this paper, it is shown that some improvements on Hilbert's inequality for double series can be established by means of Schwarz's inequality; Hilbert-Ingham's inequality can be sharpened, Hardy-Littewood's inequality and Fejer-Riesz's inequality can be refined.