- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 861948, 6 pages

http://dx.doi.org/10.1155/2013/861948

## Inequalities Similar to Hilbert's Inequality

Department of Mathematics, China Jiliang University, Hangzhou 310018, China

Received 23 June 2013; Accepted 4 August 2013

Academic Editor: Wenchang Sun

Copyright © 2013 Chang-Jian Zhao. 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.

#### Abstract

In the present paper, we establish some new inequalities similar to Hilbert’s type inequalities. Our results provide some new estimates to these types of inequalities.

#### 1. Introduction

The well-known classical Hilbert’s double-series inequality can be stated as follows [1, page 253].

Theorem A. *If such that and , where, as usual, and are the conjugate exponents of and , respectively, then
**
where depends on and only.*

In recent years, several authors [1–18] have given considerable attention to Hilbert’s double-series inequality together with its integral version, inverse version, and various generalizations. In particular, Pachpatte [11] established an inequality similar to inequality (1) as follows.

Theorem 1. *Let be constant and . If and are real-valued functions defined for and , respectively, and . Moreover, define the operators by . Then,
*

The first aim of this paper is to establish a new inequality similar to Hilbert’s type inequality. Our result provides new estimates to this type of inequality.

Theorem 2. *Let be constants, and . For , let be real-valued functions defined for , where ; , and let , be natural numbers. Let , and define the operators by
**
Then,
**
where
*

*Remark 3. *Inequality (4) is just a similar version of the following inequality established by Pachpatte [11]:

On the other hand, let and change to and , respectively, and, with appropriate transformation, we have where and is as in (6). This is just a similar version of inequality (2) in Theorem 1.

The integral analogue of inequality (1) in Theorem A is as follows [1, page 254].

Theorem B. *Let , , , , and be as in Theorem A. If and , then
**
where depends on and only.*

In [11], Pachpatte also established a similar version of inequality (10) as follows.

Theorem 4. *Let be constants, and . If and are real-valued continuous functions defined on and , respectively, and let . Then,
*

Another aim of this paper is to establish a new integral inequality similar to Hilbert’s type inequality.

Theorem 5. *Let , and . For , let , be real-valued differentiable functions defined on , where , and . As usual, partial derivatives of are denoted by , and so forth. Let
**
Then,**where
**
and is as in (6).*

*Remark 6. *Inequality (13) is just a similar version of the following inequality established by Pachpatte [11]:

On the other hand, let and change to and , respectively, and, with appropriate transformation, we have where

This is just a similar version of inequality (11) in Theorem 4.

#### 2. Proof of Theorems

*Proof of Theorem 2. *From the hypotheses of Theorem 2, we have
By using Hölder’s inequality and noticing the reverse Young’s inequality [19],
for positive real numbers , and , , where is as in (6). Hence,
Dividing both sides of (20) by
taking the sum of both sides of (20) over and from 1 to and , respectively, and making use of Hölder’s inequality, we have

This completes the proof.

*Proof of Theorem 5. *From the hypotheses of Theorem 5, we obtain for :
From (23), Hölder’s integral inequality and in view of the reverse Young’s inequality (19), we have
Integrating both sides of (24) over and from to and (), respectively, and by using Hölder’s integral inequality, we arrive at

This completes the proof.

#### Acknowledgment

The research is supported by the National Natural Science Foundation of China (11371334).

#### References

- G. H. Hardy, J. E. Littlewood, and G. Pólya,
*Inequalities*, Cambridge University Press, Cambridge, Mass, USA, 1934. - B. G. Pachpatte, “On some new inequalities similar to Hilbert's inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 226, no. 1, pp. 166–179, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. D. Handley, J. J. Koliha, and J. E. Pečarić, “New Hilbert-Pachpatte type integral inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 257, no. 1, pp. 238–250, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Gao and B. Yang, “On the extended Hilbert's inequality,”
*Proceedings of the American Mathematical Society*, vol. 126, no. 3, pp. 751–759, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Jichang, “On new extensions of Hilbert's integral inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 235, no. 2, pp. 608–614, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - B. Yang, “On new generalizations of Hilbert's inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 248, no. 1, pp. 29–40, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. J. Zhao, “On inverses of disperse and continuous Pachpatte's inequalities,”
*Acta Mathematica Sinica*, vol. 46, no. 6, pp. 1111–1116, 2003. View at MathSciNet - C. J. Zhao, “Generalization on two new Hilbert type inequalities,”
*Journal of Mathematics*, vol. 20, no. 4, pp. 413–416, 2000. View at Zentralblatt MATH · View at MathSciNet - C. J. Zhao and L. Debnath, “Some new inverse type Hilbert integral inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 262, no. 1, pp. 411–418, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. D. Handley, J. J. Koliha, and J. Pečarić, “A Hilbert type inequality,”
*Tamkang Journal of Mathematics*, vol. 31, no. 4, pp. 311–315, 2000. View at Zentralblatt MATH · View at MathSciNet - B. G. Pachpatte, “Inequalities similar to certain extensions of Hilbert's inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 243, no. 2, pp. 217–227, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. F. Beckenbach and R. Bellman,
*Inequalities*, Springer, Berlin, Germany, 1961. View at MathSciNet - C. J. Zhao, J. Pecarić, and G. S. Leng, “Inverses of some new inequalities similar to Hilbert's inequalities,”
*Taiwanese Journal of Mathematics*, vol. 10, no. 3, pp. 699–712, 2006. View at Zentralblatt MATH · View at MathSciNet - S. S. Dragomir and Y.-H. Kim, “Hilbert-Pachpatte type integral inequalities and their improvement,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 4, no. 1, article 16, 2003. View at Zentralblatt MATH · View at MathSciNet - G. A. Anastassiou, “Hilbert-Pachpatte type fractional integral inequalities,”
*Mathematical and Computer Modelling*, vol. 49, no. 7-8, pp. 1539–1550, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Jin and L. Debnath, “On a Hilbert-type linear series operator and its applications,”
*Journal of Mathematical Analysis and Applications*, vol. 371, no. 2, pp. 691–704, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Yang, “A half-discrete Hilbert-type inequality with a non-homogeneous kernel and two variables,”
*Mediterranean Journal of Mathematics*, vol. 10, no. 2, pp. 677–692, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - K. Jichang and L. Debnath, “On Hilbert type inequalities with non-conjugate parameters,”
*Applied Mathematics Letters*, vol. 22, no. 5, pp. 813–818, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Tominaga, “Specht's ratio in the Young inequality,”
*Scientiae Mathematicae Japonicae*, vol. 55, no. 3, pp. 583–588, 2002. View at Zentralblatt MATH · View at MathSciNet