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# On Hilbert-Pachpatte Multiple Integral Inequalities

*Journal of Inequalities and Applications*
**volumeÂ 2010**, ArticleÂ number:Â 820857 (2010)

## 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

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 ,

Proof.

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

Let us note the following means inequality:

We obtain that

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

The proof is complete.

Remark 2.2.

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

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

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

where

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

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

This completes the proof of Theorem 2.3.

Remark 2.4.

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

where

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

where

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

Proof.

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

Hence

Remark 2.6.

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

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

Remark 2.7.

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

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

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## 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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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

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DOI: https://doi.org/10.1155/2010/820857