- Research Article
- Open Access
A Hilbert's Inequality with a Best Constant Factor
© Z. Zeng and Z.-t. Xie. 2009
- Received: 6 February 2009
- Accepted: 23 July 2009
- Published: 23 August 2009
We give a new Hilbert's inequality with a best constant factor and some parameters.
- Constant Factor
- Weight Coefficient
- Equivalent Form
- Strict Inequality
- Integral Inequality
where the constant factors are all the best possible . It attracted some attention in the recent years. Actually, inequalities (1.1) and (1.2) have many generalizations and variants. Equation (1.1) has been strengthened by Yang and others ( including integral inequalities ) [2–11].
In 2006, Yang gave an extension of  as follows.
In 2007 Xie gave a new Hilbert-type Inequality  as follows.
If , and the right of the following inequalities converges to some positive numbers, then
The main objective of this paper is to build a new Hilbert's inequality with a best constant factor and some parameters.
In the following, we always suppose that
(1) , , ,
(2)both functions and are differentiable and strict increasing in and respectively,
(3) are strictly increasing in and respectively. is strict decreasing on and ,
On the other hand, . Setting , then Similarly, .
The lemma is proved.
There for any ).
Easily, had up bounded when
as . By (3.2), (3.5) taking the form of strict inequality, we have (3.1).
If , then both constant factors, and of (3.1) and (3.2), are the best possible.
We have , . This contracts the fact that .
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1952:xii+324.MATHGoogle Scholar
- Yang BC: On Hilbert's inequality with some parameters. Acta Mathematica Sinica. Chinese Series 2006,49(5):1121–1126.MathSciNetMATHGoogle Scholar
- Xie Z: A new Hilbert-type inequality with the kernel of --homogeneous. Journal of Jilin University. Science Edition 2007,45(3):369–373.MathSciNetMATHGoogle Scholar
- Xie Z, Yang B: A new Hilbert-type integral inequality with some parameters and its reverse. Kyungpook Mathematical Journal 2008,48(1):93–100.MathSciNetView ArticleMATHGoogle Scholar
- Yang B: A Hilbert-type inequality with a mixed kernel and extensions. Journal of Sichuan Normal University. Natural Science 2008,31(3):281–284.MATHGoogle Scholar
- Xie Z, Zeng Z: A Hilbert-type integral with parameters. Journal of Xiangtan University. Natural Science 2007,29(3):24–28.MATHGoogle Scholar
- Wenjie W, Leping H, Tieling C: On an improvenment of Hardy-Hilbert's type inequality with some parameters. Journal of Xiangtan University. Natural Science 2008,30(2):12–14.Google Scholar
- 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.2008.02.007MathSciNetView ArticleMATHGoogle Scholar
- Yang B: On an extended Hardy-Hilbert's inequality and some reversed form. International Mathematical Forum 2006,1(37–40):1905–1912.MathSciNetMATHGoogle Scholar
- Xie Z: A Hilbert-type inequality with the kernel of irrational expression. Mathematics in Practice and Theory 2008,38(16):128–133.MathSciNetGoogle Scholar
- Xie Z, Rong JM: A new Hilbert-type inequality with some parameters. Journal of South China Normal University. Natural Science Edition 2008,120(2):38–42.MathSciNetMATHGoogle Scholar
- Kang J: Applied Inequalities. Shangdong Science and Technology 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.