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

- Zheng Zeng
^{1}and - Zitian Xie
^{2}Email author

**2010**:256796

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

© Zheng Zeng and Zitian Xie. 2010

**Received: **5 May 2010

**Accepted: **14 July 2010

**Published: **2 August 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.

## Keywords

## 1. Introduction

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].

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.

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.

## 2. Some Lemmas

## 3. Main Results

Theorem 3.1.

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

Proof.

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).

Using (3.2), we have (3.1).

Inequalities (3.1) and (3.2) are equivalent.

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.

## Authors’ Affiliations

## References

- Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. Cambridge University Press, London, UK; 1952.MATHGoogle Scholar - Hardy GH: Note on a theorem of Hilbert concerning series of positive terms.
*Proceedings of the London Mathematical Society*1925, 23(2):45–46.Google Scholar - Mitrinović DS, Pečarić JE, Fink AM:
*Inequalities Involving Functions and Their Integrals and Derivatives*.*Volume 53*. Kluwer Academic, Boston, Mass, USA; 1991:xvi+587.View ArticleMATHGoogle Scholar - 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.2007.06.059MathSciNetView ArticleMATHGoogle Scholar - Xie 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.MathSciNetMATHGoogle Scholar - Xie Z, Zeng Z: The Hilbert-type integral inequality with the system kernel of - degree homogeneous form.
*Kyungpook Mathematical Journal*2010, 50: 297–306.MathSciNetView ArticleMATHGoogle Scholar - Yang B: A new Hilbert-type integral inequality with some parameters.
*Journal of Jilin University*2008, 46(6):1085–1090.MathSciNetGoogle Scholar - Xie Z, Yang B: A new Hilbert-type integral inequality with some parameters and its reverse.
*Kyungpook Mathematical Journal*2008, 48(1):93–100.MathSciNetView ArticleMATHGoogle Scholar - Xie Z: A new Hilbert-type inequality with the kernel of --homogeneous.
*Journal of Jilin University*2007, 45(3):369–373.MathSciNetMATHGoogle Scholar - Xie Z, Murong J: A reverse Hilbert-type inequality with some parameters.
*Journal of Jilin University*2008, 46(4):665–669.MathSciNetMATHGoogle Scholar - 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.2008.02.007MathSciNetView ArticleMATHGoogle Scholar - Yang B: A Hilbert-type inequality with a mixed kernel and extensions.
*Journal of Sichuan Normal University*2008, 31(3):281–284.MATHGoogle Scholar - Xie Z, Zeng Z: A Hilbert-type inequality with parameters.
*Natural Science Journal of Xiangtan University*2007, 29(3):24–28.MATHGoogle Scholar - Zeng Z, Xie Z: A Hilbert's inequality with a best constant factor.
*Journal of Inequalities and Applications*2009, 2009:-8.Google Scholar - Yang B: A bilinear inequality with a -order homogeneous kernel.
*Journal of Xiamen University*2006, 45(6):752–755.MathSciNetMATHGoogle Scholar - Yang B: On Hilbert's inequality with some parameters.
*Acta Mathematica Sinica*2006, 49(5):1121–1126.MathSciNetMATHGoogle Scholar - Brnetić I, Pečarić J: Generalization of Hilbert's integral inequality.
*Mathematical Inequalities and Application*2004, 7(2):199–205.View ArticleMATHGoogle Scholar - 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/S0004972700038454MathSciNetView ArticleMATHGoogle Scholar - Xie 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.MathSciNetMATHGoogle Scholar - Xie Z, Liu X: A new Hilbert-type integral inequality and its reverse.
*Journal of Henan University*2009, 39(1):10–13.MATHGoogle Scholar - Xie Z, Fu BL: A new Hilbert-type integral inequality with a best constant factor.
*Journal of Wuhan University*2009, 55(6):637–640.MathSciNetGoogle Scholar - Kang J:
*Applied Inequalities*. Shangdong Science and Technology Press, Jinan, China; 2004.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.