- Research Article
- Open Access

# A New Hilbert-Type Linear Operator with a Composite Kernel and Its Applications

- Wuyi Zhong
^{1}Email author

**2010**:393025

https://doi.org/10.1155/2010/393025

© Wuyi Zhong. 2010

**Received:**20 April 2010**Accepted:**31 October 2010**Published:**2 November 2010

## Abstract

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.

## Keywords

- Linear Operator
- Weight Function
- Kernel Function
- Differentiable Function
- Constant Factor

## 1. Introduction

In 1908, Weyl [1] published the well-known Hilbert's inequality as follows:

where the constant factor is the best possible.

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 [5] gave a more accurate form of (1.3).

where is a linear operator, . is the norm of the sequence with a weight function . is a formal inner product of the sequences and .

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 [6] is introduced.

## 2. Lemmas

Lemma 2.1.

Then, it has the following.

- (2)

- (3)

The expression (2.8) holds, and Lemma 2.1 is proved.

Lemma 2.2.

Then, it has

- (1)
Letting , , it can be proved that satisfy (2.19) as in [5]. Similarly, it can be shown that satisfy (2.19) also.

- (2)

It means that (2.21) holds. The proof for Lemma 2.2 is finished.

## 3. Main Results

- (1)

(2) set

*the norm of the sequence with a weight function*. Similarly,

*the real spaces of sequences*

*,*and

*the norm*can be defined as well,

- (3)
define

*a Hilbert-type linear operator*, for all ,

- (4)
for all , , define

*the formal inner product of**and*as

define two *weight coefficients*
and
as

Then it has some results in the following theorems.

Theorem 3.1.

then for all and , it has the following:

It means that ,

(2) is a bounded linear operator and

where , are defined by (3.4), is defined as (3.3).

Proof.

This means that , , and . is a bounded linear operator.

In view of (3.13) and (3.18), letting , it has . This means that ; that is, . Theorem 3.1 is proved.

Theorem 3.2.

then it has

(2)if and , then

where inequality (3.27) is equivalent to (3.26) and the constant factor is the best possible.

Proof.

(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.

## 4. Applications

Example 4.1.

Set , be two pairs of conjugate exponents and , , , . Then it has the following.

(2)If , then

where the constant factors and are both the best possible. Inequality (4.2) is equivalent to (4.1).

Proof.

When and ; that is, , and , , by Theorem 3.2, inequality (4.1) holds, so does (4.2). And (4.2) is equivalent to (4.1), and the constant factors and are both the best possible.

Example 4.2.

Set , be two pairs of conjugate exponents and , , , . Then it has the following.

(2)If , then

where inequality (4.8) is equivalent to (4.7) and the constant factors and are both the best possible.

Proof.

When and ; that is, , and , , by Theorem 3.2, inequality (4.7) holds, so does (4.8). And (4.8) is equivalent to (4.7), and the constant factors and are both the best possible.

Remark 4.3.

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.

## Declarations

### Acknowledgment

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.

## Authors’ Affiliations

## References

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*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

## Copyright

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.