- Research Article
- Open Access
On a Multiple Hilbert's Inequality with Parameters
© Qiliang Huang. 2010
Received: 12 May 2010
Accepted: 31 August 2010
Published: 2 September 2010
By introducing multiparameters and conjugate exponents and using Hadamard's inequality and the way of real analysis, we estimate the weight coefficients and give a multiple more accurate Hilbert's inequality, which is an extension of some published results. We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.
where the constant factor is the best possible. Inequalities (1.1)–(1.4) are important in analysis and their applications (cf. ). In near one century, there are many improvements, generalizations and, applications of (1.1)–(1.4) in numerous literatures and monographs of mathematics (cf. [2–18]). Yang and Huang also considered the multiple Hilbert-type integral inequality (cf. [19, 20]). Recently, Yang summarized the methods of introducing parameters and estimating the weight coefficients to extend Hilbert-type inequalities for the past 100 years. Some representative results are as follows (cf. [21, 22]):
The constant factors in the above five inequalities are all the best possible. Inequalities (1.5) and (1.7) are generalizations of inequality (1.2), and inequality (1.9) is a multiple extension of (1.1). Inequalities (1.6) and (1.8) are the equivalent forms of (1.5) and (1.7), which are extensions of (1.4).
In this paper, by introducing multi-parameters and conjugate exponents and using Hadamard's inequality, we estimate the weight coefficients and give a multiple more accurate Hilbert 's inequality, which is an extension of inequalities (1.5), (1.7), and (1.9). We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.
2. Some Lemmas
and then (2.1) is valid.
Hence, we prove that the left-hand side of (2.3) is valid.
and then (2.11) is valid by using inequality (2.3).
Hence, (2.12) is valid.
3. Main Results
Then (3.3) is valid, which is equivalent to (3.2).
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).
- Weyl H: Singulare integral gleichungen mit besonderer berucksichtigung des fourierschen integral theorems, Inaugural-Dissertation. Göttingen University, Göttingen, Germany; 1908.Google Scholar
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.MATHGoogle 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
- Zhong W: A Hilbert-type linear operator with the norm and its applications. Journal of Inequalities and Applications 2009, 2009:-18.Google Scholar
- Kuang JC: Applied Inequalitie. Shangdong Science Technic Press, Jinan, China; 2004.Google Scholar
- Hu K: Some Problems in Analysis Inequalities. Wuhan University Press, Wuhan, China; 2007.Google Scholar
- Magnus W: On the spectrum of Hilbert's matrix. American Journal of Mathematics 1950, 72: 699–704. 10.2307/2372284MathSciNetView ArticleMATHGoogle Scholar
- Yang BC, Gao MZ: On a best value of Hardy-Hilbert's inequality. Advances in Mathematics 1997, 26(2):159–164.MathSciNetMATHGoogle Scholar
- Gao MZ, Yang BC: On the extended Hilbert's inequality. Proceedings of the American Mathematical Society 1998, 126(3):751–759. 10.1090/S0002-9939-98-04444-XMathSciNetView ArticleMATHGoogle Scholar
- Jichang K: On new extensions of Hilbert's integral inequality. Journal of Mathematical Analysis and Applications 1999, 235(2):608–614. 10.1006/jmaa.1999.6373MathSciNetView ArticleMATHGoogle Scholar
- Yang BC, Debnath L: On the extended Hardy-Hilbert's inequality. Journal of Mathematical Analysis and Applications 2002, 272(1):187–199. 10.1016/S0022-247X(02)00151-8MathSciNetView ArticleMATHGoogle Scholar
- Yang BC: An extension of Hardy-Hilbert's inequality. Chinese Annals of Mathematics 2002, 23(2):247–254.MathSciNetMATHGoogle Scholar
- Yang BC, Rassias TM: On a new extension of Hilbert's inequality. Mathematical Inequalities & Applications 2005, 8(4):575–582.MathSciNetView ArticleMATHGoogle Scholar
- Yang B: On a new extension of Hilbert's inequality with some parameters. Acta Mathematica Hungarica 2005, 108(4):337–350. 10.1007/s10474-005-0229-4MathSciNetView ArticleMATHGoogle Scholar
- Yang BC: Hilbert's inequality with some parameters. Acta Mathematica Sinica. Chinese Series 2006, 49(5):1121–1126.MathSciNetMATHGoogle Scholar
- Yang BC: A dual Hardy-Hilbert's inequality and generalizations. Advances in Mathematics 2006, 35(1):102–108.MathSciNetGoogle Scholar
- Yang BC: On a Hilbert-type operator with a symmetric homogeneous kernel of --order and applications. Journal of Inequalities and Applications 2007, -9.Google Scholar
- Yang BC: On the norm of a linear operator and its applications. Indian Journal of Pure and Applied Mathematics 2008, 39(3):237–250.MathSciNetMATHGoogle Scholar
- Yang BC: Hilbert-type Integral Inequalities. Bentham Science, Oak Park, Ill, USA; 2009.Google Scholar
- Huang Q, Yang BC: On a multiple Hilbert-type integral operator and applications. Journal of Inequalities and Applications 2009, 2009:-13.Google Scholar
- Yang BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing, China; 2009.Google Scholar
- Yang BC: A survey of the study of Hilbert-type inequalities with parameters. Advances in Mathematics 2009, 38(3):257–268.MathSciNetGoogle 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.