# The Hilbert-Type Integral Inequality with the System Kernel of-λ Degree Homogeneous Form

Xie, Zitian;Zeng, Zheng

• Accepted : 2010.01.28
• Published : 2010.06.30
• 18 4

#### Abstract

In this paper, the integral operator is used. We give a new Hilbert-type integral inequality, whose kernel is the homogeneous form with degree - $\lambda$ and with three pairs of conjugate exponents and the best constant factor and its reverse form are also derived. It is shown that the results of this paper represent an extension as well as some improvements of the earlier results.

#### Keywords

Integral operator;Hilbert-type integral inequality;weight function;H$\"{o}$lder's inequality

#### References

1. Hardy G. H., Littlewood J. E. and Polya G., Inequalities, Cambridge University Press, Cambridge, 1952.
2. Bicheng Yang, On the the norm of an intergal operator and applications, J. Math. Anal. Appl., 2006, 321: 182-192. https://doi.org/10.1016/j.jmaa.2005.07.071
3. Zitian Xie and Zeng Zheng, A Hilbert-type integral inequality whose kernel is a ho- mogeneous form of degree-3, J.Math.Appl., 2008, (339): 324-331. https://doi.org/10.1016/j.jmaa.2007.06.059
4. Zitian Xie and Zeng Zheng, A Hilbert-type inequality with parameters, Natural science Journal of Xiangtan University, 2007, 29(3): 24-28.
5. Bicheng Yang, A Hilbert-type inequality with a mixed kemel and extensions, Journal of Sichuan Normal University (Natural Science), 2008, 31(3): 281-284.
6. Zitian Xie and Fang min Zhou, A Generalization of a Hilbert-type inequality with a best constant factor, Journal of Sichuan Normal University (Natural Science), 2009, 32(5): 626-629.
7. Zitian Xie and Benlu Fu, A new Hilbert-type integral inequality with a best constant factor, J. Wuhan Univ. (Nat.Sci.Ed), 2009, 55(6): 637-640.
8. Zitian Xie and Xingdong Liu, A new Hilbert-type integral inequality and its reverse, Journal of Henan University (Science Edition), 2009, 39(1)10-13.
9. Zitian Xie and Bicheng Yang, A new Hilbert-type integral inequality with some pa- rameters and its reverse, Kyungpook Mathe.J., 2008, (48): 93-100.
10. Bicheng Yang, On a Hilbert-type integral inequality with multiple-parameters, Journal of Southwest China Normal University(Natural Science), 2007, 32(5): 33-38.
11. Bicheng Yang, A relation to Hilbert's integral inequality and some base Hilbert-type inequalities, Journal of inequalities in pure and applied mathematics, 2008, 9(2), Article 59: 1-8.
12. Bicheng Yang, A base Hilbert-type integral inequality with the homogeneous kernel of -1-degree and extensions, Journal of Guangong Education Institute, 2008, 28(3): 5-10.
13. Dongmei Xin, On a new Hilbert-type integral inequality, Kyungpook Math.J., 2009, (49): 393-401.
14. Wuyi Zhong and Bicheng Yang, A reverse Hilbert's type integral inequality with some parameters and the equalent forms, Pure and Applied Mathematics, 2008, 24(2), 401-407.
15. Bicheng Yang, On a Hilbert-type integral inequality with a parameter, Pure and Applied Mathematics, 2008, 24(3), 489-494.
16. Bicheng Yang, A new Hilbert-type integral inequality, Journal of Jilin University (Science Edition) 2007, 45(1): 63-67.
17. Bicheng Yang, A Hilbert-type integral inequality, Journal of Zhejiang University (Science Edition), 2007, 34(2): 121-124.
18. Bicheng Yang, An extension of the Hilbert's type integral inequality and its applica- tions , J. of Math., (PRC)2007, 27(3): 285-290.
19. Bicheng Yang, On a reverse Hardy-Hilbert's inequality ,Kyungpook Mathe.J., 2007, (47): 411-423.
20. Zitian Xie and Ju-min Mu Rong, A New Hilbert type Inequality with some parameters, Journal of South China Normal University (Natural Science Edition) 2008, 120(2): 38-42.
21. Zitian Xie, A reverse Hilbert-type inequality with a best constant Factor, J. Math. Anal. Appl., 2008, 343: 1154-1160. https://doi.org/10.1016/j.jmaa.2008.02.007
22. Jichang Kang, Applied Inequalities, Shangdong Science and Technology press, Jinan, 2004.

#### Cited by

1. A New Hilbert-type Inequality with the Integral in Whole Plane vol.52, pp.3, 2012, https://doi.org/10.5666/KMJ.2012.52.3.291
2. vol.02, pp.01, 2012, https://doi.org/10.12677/pm.2012.21003
3. A Hilbert-Type Inequality with Some Parameters and the Integral in Whole Plane vol.01, pp.03, 2011, https://doi.org/10.4236/apm.2011.13019