Advertisement

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

  • Zheng Zeng
  • Zitian Xie
Open Access
Research Article

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

Weight Function Measurable Function Mathematical Analysis Complex Analysis Constant Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

where the constant factor Open image in new window 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 Open image in new window , Open image in new window , Open image in new window such that Open image in new window and Open image in new window then we have the following Hardy-Hilbert's integral inequality:

where the constant factor Open image in new window 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, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

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

where the constant factor Open image in new window 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: Open image in new window Open image in new window , Open image in new window

2. Some Lemmas

We start by introducing some lemmas.

Lemma 2.1.

Proof.

The lemma is proved.

Lemma 2.2.

Define the weight functions as follow:

then Open image in new window .

Proof.

We only prove that Open image in new window for Open image in new window .

and the lemma is proved.

Lemma 2.3.

Proof.

Easily, we get the following:
we have that Open image in new window is an even function on Open image in new window , then

where Open image in new window and we have Open image in new window

Similarly, Open image in new window The lemma is proved.

Lemma 2.4.

If Open image in new window is a nonnegative measurable function and Open image in new window , then

Proof.

By Lemma 2.2, we find that

3. Main Results

Theorem 3.1.

If both functions, Open image in new window and Open image in new window , are nonnegative measurable functions and satisfy Open image in new window and Open image in new window , then

Inequalities (3.1) and (3.2) are equivalent, and where the constant factors Open image in new window and Open image in new window are the best possibles.

Proof.

If (2.13) takes the form of equality for some Open image in new window , then there exists constants Open image in new window and Open image in new window , such that they are not all zero, and
Hence, there exists a constant Open image in new window , such that

We claim that Open image in new window . In fact, if Open image in new window , then Open image in new window a.e. in Open image in new window which contradicts the fact that Open image in new window . In the same way, we claim that Open image in new window 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).

Inequalities (3.1) and (3.2) are equivalent.

If the constant factor Open image in new window in (3.1) is not the best possible, then there exists a positive Open image in new window (with Open image in new window ), such that
For Open image in new window , by (3.7), using Lemma 2.3, we have

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

Thus we complete the proof of the theorem.

Remark 3.2.

For Open image in new window in (3.1), we have the following particular result:

References

  1. [1]
    Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, London, UK; 1952.MATHGoogle Scholar
  2. [2]
    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
  3. [3]
    Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Volume 53. Kluwer Academic, Boston, Mass, USA; 1991:xvi+587.CrossRefMATHGoogle Scholar
  4. [4]
    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.2007.06.059MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    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
  6. [6]
    Xie Z, Zeng Z: The Hilbert-type integral inequality with the system kernel of - degree homogeneous form. Kyungpook Mathematical Journal 2010, 50: 297–306.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Yang B: A new Hilbert-type integral inequality with some parameters. Journal of Jilin University 2008, 46(6):1085–1090.MathSciNetGoogle Scholar
  8. [8]
    Xie Z, Yang B: A new Hilbert-type integral inequality with some parameters and its reverse. Kyungpook Mathematical Journal 2008, 48(1):93–100.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Xie Z: A new Hilbert-type inequality with the kernel of --homogeneous. Journal of Jilin University 2007, 45(3):369–373.MathSciNetMATHGoogle Scholar
  10. [10]
    Xie Z, Murong J: A reverse Hilbert-type inequality with some parameters. Journal of Jilin University 2008, 46(4):665–669.MathSciNetMATHGoogle Scholar
  11. [11]
    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.2008.02.007MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Yang B: A Hilbert-type inequality with a mixed kernel and extensions. Journal of Sichuan Normal University 2008, 31(3):281–284.MATHGoogle Scholar
  13. [13]
    Xie Z, Zeng Z: A Hilbert-type inequality with parameters. Natural Science Journal of Xiangtan University 2007, 29(3):24–28.MATHGoogle Scholar
  14. [14]
    Zeng Z, Xie Z: A Hilbert's inequality with a best constant factor. Journal of Inequalities and Applications 2009, 2009:-8.Google Scholar
  15. [15]
    Yang B: A bilinear inequality with a -order homogeneous kernel. Journal of Xiamen University 2006, 45(6):752–755.MathSciNetMATHGoogle Scholar
  16. [16]
    Yang B: On Hilbert's inequality with some parameters. Acta Mathematica Sinica 2006, 49(5):1121–1126.MathSciNetMATHGoogle Scholar
  17. [17]
    Brnetić I, Pečarić J: Generalization of Hilbert's integral inequality. Mathematical Inequalities and Application 2004, 7(2):199–205.CrossRefMATHGoogle Scholar
  18. [18]
    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/S0004972700038454MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    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
  20. [20]
    Xie Z, Liu X: A new Hilbert-type integral inequality and its reverse. Journal of Henan University 2009, 39(1):10–13.MATHGoogle Scholar
  21. [21]
    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
  22. [22]
    Kang J: Applied Inequalities. Shangdong Science and Technology Press, Jinan, China; 2004.Google Scholar

Copyright information

© Zheng Zeng and Zitian Xie. 2010

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.

Authors and Affiliations

  1. 1.Department of MathematicsShaoguan UniversityShaoguanChina
  2. 2.Department of MathematicsZhaoqing UniversityZhaoqingChina

Personalised recommendations