A New Hilbert-Type Linear Operator with a Composite Kernel and Its Applications
© Wuyi Zhong. 2010
Received: 20 April 2010
Accepted: 31 October 2010
Published: 2 November 2010
A new Hilbert-type linear operator with a composite kernel function is built. As the applications, two new more accurate operator inequalities and their equivalent forms are deduced. The constant factors in these inequalities are proved to be the best possible.
In 1908, Weyl  published the well-known Hilbert's inequality as follows:
where the constant factors and are the best possible also. Expression (1.2) is called a more accurate form of(1.1). Some more accurate inequalities were considered by [3–5]. In 2009, Zhong  gave a more accurate form of (1.3).
By setting two monotonic increasing functions and , a new Hilbert-type inequality, which is with a composite kernel function , and its equivalent are built in this paper. As the applications, two new more accurate Hilbert-type inequalities incorporating the linear operator and the norm are deduced.
Firstly, the improved Euler-Maclaurin's summation formula  is introduced.
Then, it has the following.
The expression (2.8) holds, and Lemma 2.1 is proved.
Then, it has
Letting , , it can be proved that satisfy (2.19) as in . Similarly, it can be shown that satisfy (2.19) also.
It means that (2.21) holds. The proof for Lemma 2.2 is finished.
3. Main Results
Then it has some results in the following theorems.
then it has
(3.26) holds also.
Letting in (3.37), it has , and it means that and . Therefore, the inequality (3.36) keeps the form of the strict inequality when . In view of , inequality (3.27) holds and (3.27) is equivalent to (3.26). By , it is obvious that the constant factor is the best possible. This completes the proof of Theorem 3.2.
It can be proved similarly that, if the conditions " " in Lemma 2.1 and " " in Lemma 2.2 are changed into " " and " ", respectively, Lemmas 2.1 and 2.2 are also valid. So the conditions " " in Example 4.1 and " " in Example 4.2 can be replaced by " , " and " , ", respectively.
This paper is supported by the National Natural Science Foundation of China (no. 10871073). The author would like to thank the anonymous referee for his or her suggestions and corrections.
- Weyl H: Singulare Integralgleichungen mit besonderer Beriicksichtigung des Fourierschen Integraltheorems, Inaugeral dissertation. University of Göttingen, Göttingen, Germany; 1908.MATHGoogle Scholar
- Hardy G, Littiewood J, Polya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.Google Scholar
- Yang BC: On a more accurate Hardy-Hilbert-type inequality and its applications. Acta Mathematica Sinica 2006, 49(2):363–368.MathSciNetMATHGoogle Scholar
- Yang B: On a more accurate Hilbert's type inequality. International Mathematical Forum 2007, 2(37–40):1831–1837.MathSciNetMATHGoogle Scholar
- Zhong W: A Hilbert-type linear operator with the norm and its applications. Journal of Inequalities and Applications 2009, 2009:-18.Google Scholar
- Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing, China; 2009.Google Scholar
- Kuang 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.