- Research Article
- Open access
- Published:
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.2007.06.059
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.
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.2008.02.007
Yang 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/S0004972700038454
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.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/256796