# On Hilbert-Pachpatte Multiple Integral Inequalities

- Changjian Zhao
^{1}Email author, - Lian-ying Chen
^{1}and - Wing-Sum Cheung
^{2}

**2010**:820857

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

© Changjian Zhao et al. 2010

**Received: **11 March 2010

**Accepted: **28 July 2010

**Published: **10 August 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.

In recent years, considerable attention has been given to various extensions and improvements of the Hilbert inequality form different viewpoints [2–10]. 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.

Proof.

The proof is complete.

Remark 2.2.

Theorem 2.3.

Proof.

This completes the proof of Theorem 2.3.

Remark 2.4.

Theorem 2.5.

Proof.

Remark 2.6.

Remark 2.7.

## Declarations

### 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.

## Authors’ Affiliations

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## 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.