- Research Article
- Open Access
On a Multiple Hilbert-Type Integral Operator and Applications
© Q. Huang and B. Yang. 2010
- Received: 19 August 2009
- Accepted: 9 December 2009
- Published: 12 January 2010
By using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral operator with the homogeneous kernel of -degree ( ) and its norm are considered. As for applications, two equivalent inequalities with the best constant factors, the reverses, and some particular norms are obtained.
- Weight Function
- Measurable Function
- Integral Operator
- Convergence Theorem
- Equivalent Form
where the constant factor is the best possible. Define the Hardy-Hilbert's integral operator as follows: for Then in view of (1.2), it follows that and Since the constant factor in (1.2) is the best possible, we find that (cf.)
where the constant factor is the best possible. For , and in (1.6), we obtain (1.3) and (1.4). Inequality (1.6) is some extensions of the results in [6, 8–11]. In 2006, reference  also considered a multiple Hilbert-type integral operator with the homogeneous kernel of -degree and its inequality with the norm, which is the best extension of (1.2).
In this paper, by using the way of weight functions and the technic of real analysis, a new multiple Hilbert-type integral operator with the norm is considered, which is an extension of the result in . As for applications, an extended multiple Hilbert-type integral inequality and the equivalent form, the reverses, and some particular norms are obtained.
and then (2.1) is valid.
Then by combination with (2.15), we have (2.10).
and then (3.6) is valid, which is equivalent to (3.7).
and Hence is the best value of (3.7). We conform that the constant factor in (3.6) is the best possible; otherwise, we can get a contradiction by (3.9) that the constant factor in (3.7) is not the best possible. Therefore
This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026), and Guangdong Natural Science Foundation (no. 7004344).
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.MATHGoogle Scholar
- Carleman T: Sur les Equations Integrals Singulieres a Noyau Reed et Symetrique. Volume 923. Uppsala universitet, Uppsala, Sweden;Google Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications. Volume 53. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.View ArticleMATHGoogle Scholar
- Zhang K: A bilinear inequality. Journal of Mathematical Analysis and Applications 2002,271(1):288–296. 10.1016/S0022-247X(02)00104-XMathSciNetView ArticleMATHGoogle Scholar
- Yang B: On a extansion of Hilbert's integral inequality with some parameters. The Australian Journal of Mathematical Analysis and Applications 2004,1(1, article 11):1–8.Google Scholar
- Yang B, Brnetić I, Krnić M, Pečarić J: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Mathematical Inequalities & Applications 2005,8(2):259–272.MathSciNetMATHGoogle Scholar
- Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijin, China; 2009.Google Scholar
- Hong Y: All-sided generalization about Hardy-Hilbert integral inequalities. Acta Mathematica Sinica 2001,44(4):619–625.MathSciNetMATHGoogle Scholar
- He L, Yu J, Gao M: An extension of the Hilbert's integral inequality. Journal of Shaoguan University ( Natural Science) 2002,23(3):25–30.Google Scholar
- Yang B: A multiple Hardy-Hilbert integral inequality. Chinese Annals of Mathematics. Series A 2003,24(6):25–30.MathSciNetGoogle Scholar
- Yang B, Rassias ThM: On the way of weight coefficient and research for the Hilbert-type inequalities. Mathematical Inequalities & Applications 2003,6(4):625–658.MathSciNetView ArticleMATHGoogle Scholar
- Bényi Á, Choonghong O: Best constants for certain multilinear integral operators. Journal of Inequalities and Applications 2006, 2006:-12.Google Scholar
- Kuang J: Introduction to Real Analysis. Hunan Education Press, Chansha, China; 1996.Google Scholar
- Kuang J: Applied Inequalities. Shangdong Science Technic Press, Jinan, China; 2004.Google Scholar
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.