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# On Hilbert-Pachpatte Multiple Integral Inequalities

## Abstract

We establish some multiple integral Hilbert-Pachpatte-type inequalities. As applications, we get some inverse forms of Pachpatte's inequalities which were established in 1998.

## 1. Introduction

In 1934, Hilbert [1] established the following well-known integral inequality.

If , , , and , then

(1.1)

where is the best value.

In recent years, considerable attention has been given to various extensions and improvements of the Hilbert inequality form different viewpoints [2â€“10]. In particular, Pachpatte [11] proved some inequalities similar to Hilbert's integral inequalities in 1998. In this paper, we establish some new multiple integral Hilbert-Pachpatte-type inequalities.

## 2. Main Results

Theorem 2.1.

Let , let , , where are positive real numbers, and define , for . Then for , and ,

(2.1)

Proof.

From the hypotheses and in view of inverse HÃ¶lder integral inequality (see [12]), it is easy to observe that

(2.2)

Let us note the following means inequality:

(2.3)

We obtain that

(2.4)

Integrating both sides of (2.4) over from to and using the special case of inverse HÃ¶lder integral inequality, we observe that

(2.5)

The proof is complete.

Remark 2.2.

Taking , to (2.1), (2.1) changes to

(2.6)

This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:

(2.7)

Theorem 2.3.

Let , , , and be as in Theorem 2.1. Let be n positive functions defined for , and define where are positive real numbers. Let be n real-valued nonnegative, concave, and super-multiplicative functions defined on . Then

(2.8)

where

(2.9)

Proof.

By using Jensen integral inequality (see [11]) and inverse HÃ¶lder integral inequality (see [12]) and noticing that are real-valued super-multiplicative functions, it is easy to observe that

(2.10)

In view of the means inequality and integrating two sides of (2.10) over from to and noticing HÃ¶lder integral inequality, we observe that

(2.11)

This completes the proof of Theorem 2.3.

Remark 2.4.

Taking , to (2.8), (2.8) changes to

(2.12)

where

(2.13)

This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:

(2.14)

where

(2.15)

Theorem 2.5.

Let , , , , and be as Theorem 2.3, and define for , where are positive real numbers. Let be real-valued, nonnegative, and concave functions on . Then

(2.16)

Proof.

From the hypotheses and by using Jensen integral inequality and the inverse HÃ¶lder integral inequality, we have

(2.17)

Hence

(2.18)

Remark 2.6.

Taking , to (2.16), (2.16) changes to

(2.19)

This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:

(2.20)

Remark 2.7.

In (2.20), if , then , . Therefore (2.20) changes to

(2.21)

This is just an inverse inequality similar to the following Inequality which was proved by Pachpatte [11]:

(2.22)

## References

1. Hardy GH, Littlewood JE, PÃ³lya G: Inequalities. Cambridge University Press, Cambridge, Mass, USA; 1952:xii+324.

2. Yang B: On a relation between Hilbert's inequality and a Hilbert-type inequality. Applied Mathematics Letters 2008, 21(5):483â€“488. 10.1016/j.aml.2007.06.001

3. Anastassiou GA: Hilbert-Pachpatte type general multivariate integral inequalities. International Journal of Applied Mathematics 2007, 20(4):549â€“573.

4. Yang BC: Hilbert's inequality with some parameters. Acta Mathematica Sinica 2006, 49(5):1121â€“1126.

5. Kuang JC, Debnath L: The general form of Hilbert's inequality and its converses. Analysis Mathematica 2005, 31(3):163â€“173. 10.1007/s10476-005-0011-4

6. Zhao C-J, Cheung W-S: Sharp integral inequalities involving high-order partial derivatives. Journal of Inequalities and Applications 2008, 2008:-10.

7. Changjian Z, PecariÄ‡ J, Gangsong L: Inverses of some new inequalities similar to Hilbert's inequalities. Taiwanese Journal of Mathematics 2006, 10(3):699â€“712.

8. Yang B: On new generalizations of Hilbert's inequality. Journal of Mathematical Analysis and Applications 2000, 248(1):29â€“40. 10.1006/jmaa.2000.6860

9. Kuang JC: On new extensions of Hilbert's integral inequality. Journal of Mathematical Analysis and Applications 1999, 235(2):608â€“614. 10.1006/jmaa.1999.6373

10. Gao M, Yang B: On the extended Hilbert's inequality. Proceedings of the American Mathematical Society 1998, 126(3):751â€“759. 10.1090/S0002-9939-98-04444-X

11. Pachpatte BG: On some new inequalities similar to Hilbert's inequality. Journal of Mathematical Analysis and Applications 1998, 226(1):166â€“179. 10.1006/jmaa.1998.6043

12. Beckenbach EF, Bellman R: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 30. Springer, Berlin, Germany; 1961:xii+198.

## Acknowledgments

This paper is supported by the National Natural Sciences Foundation of China (10971205). This paper is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and an HKU Seed Grant for Basic Research.

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Correspondence to Changjian Zhao.

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Zhao, C., Chen, Ly. & Cheung, WS. On Hilbert-Pachpatte Multiple Integral Inequalities. J Inequal Appl 2010, 820857 (2010). https://doi.org/10.1155/2010/820857