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# On a New Hilbert-Type Intergral Inequality with the Intergral in Whole Plane

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 256796 (2010)

## Abstract

By introducing some parameters and estimating the weight functions, we build a new Hilbert's inequality with the homogeneous kernel of 0 order and the integral in whole plane. The equivalent inequality and the reverse forms are considered. The best constant factor is calculated using Complex Analysis.

## 1. Introduction

If , and satisfy that and then we have [1]

where the constant factor is the best possible. Inequality (1.1) is well known as Hilbert's integral inequality, which has been extended by Hardy-Riesz as [2].

If , , such that and then we have the following Hardy-Hilbert's integral inequality:

where the constant factor also is the best possible.

Both of them are important in Mathematical Analysis and its applications [3]. It attracts some attention in recent years. Actually, inequalities (1.1) and (1.2) have many generalizations and variations. Equation (1.1) has been strengthened by Yang and others (including double series inequalities) [4–21].

In 2008, Xie and Zeng gave a new Hilbert-type Inequality [4] as follows.

If,, such that and , then

where the constant factor is the best possible.

The main purpose of this paper is to build a new Hilbert-type inequality with homogeneous kernel of degree 0, by estimating the weight function. The equivalent inequality is considered.

In the following, we always suppose that: ,

## 2. Some Lemmas

We start by introducing some lemmas.

Lemma 2.1.

If, then

Proof.

We have

Setting , then

we find that , then

The lemma is proved.

Lemma 2.2.

Define the weight functions as follow:

then .

Proof.

We only prove that for .

Using Lemma 2.1, setting and,

and the lemma is proved.

Lemma 2.3.

For and define both functions as follows:

then

Proof.

Easily, we get the following:

Let , using and

we have that is an even function on , then

Setting then

where and we have

Similarly, The lemma is proved.

Lemma 2.4.

If is a nonnegative measurable function and , then

Proof.

By Lemma 2.2, we find that

## 3. Main Results

Theorem 3.1.

If both functions,and, are nonnegative measurable functions and satisfy and , then

Inequalities (3.1) and (3.2) are equivalent, and where the constant factors and are the best possibles.

Proof.

If (2.13) takes the form of equality for some , then there exists constants and , such that they are not all zero, and

Hence, there exists a constant , such that

We claim that . In fact, if , then a.e. in which contradicts the fact that . In the same way, we claim that This is too a contradiction and hence by (2.13), we have (3.2).

By Hölder's inequality with weight [22] and (3.2), we have the following:

Using (3.2), we have (3.1).

Setting , then by (2.13), we have . If then (3.2) is proved. If by (3.1), we obtain

Inequalities (3.1) and (3.2) are equivalent.

If the constant factor in (3.1) is not the best possible, then there exists a positive (with ), such that

For , by (3.7), using Lemma 2.3, we have

Hence, we find For , it follows that , which contradicts the fact that . Hence the constant in (3.1) is the best possible.

Thus we complete the proof of the theorem.

Remark 3.2.

For in (3.1), we have the following particular result:

## References

Hardy GH, Littlewood JE, Pólya G:

*Inequalities*. Cambridge University Press, London, UK; 1952.Hardy GH: Note on a theorem of Hilbert concerning series of positive terms.

*Proceedings of the London Mathematical Society*1925, 23(2):45–46.Mitrinović DS, Pečarić JE, Fink AM:

*Inequalities Involving Functions and Their Integrals and Derivatives*.*Volume 53*. Kluwer Academic, Boston, Mass, USA; 1991:xvi+587.Xie Z, Zeng Z: A Hilbert-type integral inequality whose kernel is a homogeneous form of degree .

*Journal of Mathematical Analysis and Applications*2008, 339(1):324–331. 10.1016/j.jmaa.2007.06.059Xie Z, Zeng Z: A Hilbert-type integral inequality with a non-homogeneous form and a best constant factor.

*Advances and Applications in Mathematical Science*2010, 3(1):61–71.Xie Z, Zeng Z: The Hilbert-type integral inequality with the system kernel of - degree homogeneous form.

*Kyungpook Mathematical Journal*2010, 50: 297–306.Yang B: A new Hilbert-type integral inequality with some parameters.

*Journal of Jilin University*2008, 46(6):1085–1090.Xie Z, Yang B: A new Hilbert-type integral inequality with some parameters and its reverse.

*Kyungpook Mathematical Journal*2008, 48(1):93–100.Xie Z: A new Hilbert-type inequality with the kernel of --homogeneous.

*Journal of Jilin University*2007, 45(3):369–373.Xie Z, Murong J: A reverse Hilbert-type inequality with some parameters.

*Journal of Jilin University*2008, 46(4):665–669.Xie Z: A new reverse Hilbert-type inequality with a best constant factor.

*Journal of Mathematical Analysis and Applications*2008, 343(2):1154–1160. 10.1016/j.jmaa.2008.02.007Yang B: A Hilbert-type inequality with a mixed kernel and extensions.

*Journal of Sichuan Normal University*2008, 31(3):281–284.Xie Z, Zeng Z: A Hilbert-type inequality with parameters.

*Natural Science Journal of Xiangtan University*2007, 29(3):24–28.Zeng Z, Xie Z: A Hilbert's inequality with a best constant factor.

*Journal of Inequalities and Applications*2009, 2009:-8.Yang B: A bilinear inequality with a -order homogeneous kernel.

*Journal of Xiamen University*2006, 45(6):752–755.Yang B: On Hilbert's inequality with some parameters.

*Acta Mathematica Sinica*2006, 49(5):1121–1126.Brnetić I, Pečarić J: Generalization of Hilbert's integral inequality.

*Mathematical Inequalities and Application*2004, 7(2):199–205.Brnetić I, Krnić M, Pečarić J: Multiple Hilbert and Hardy-Hilbert inequalities with non-conjugate parameters.

*Bulletin of the Australian Mathematical Society*2005, 71(3):447–457. 10.1017/S0004972700038454Xie Z, Zhou FM: A generalization of a Hilbert-type inequality with the best constant factor.

*Journal of Sichuan Normal University*2009, 32(5):626–629.Xie Z, Liu X: A new Hilbert-type integral inequality and its reverse.

*Journal of Henan University*2009, 39(1):10–13.Xie Z, Fu BL: A new Hilbert-type integral inequality with a best constant factor.

*Journal of Wuhan University*2009, 55(6):637–640.Kang J:

*Applied Inequalities*. Shangdong Science and Technology Press, Jinan, China; 2004.

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Zeng, Z., Xie, Z. On a New Hilbert-Type Intergral Inequality with the Intergral in Whole Plane.
*J Inequal Appl* **2010**, 256796 (2010). https://doi.org/10.1155/2010/256796

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DOI: https://doi.org/10.1155/2010/256796