Research

# On Hilbert type inequalities

Chang-Jian Zhao1* and Wing-Sum Cheung2

Author Affiliations

1 Department of Mathematics, China Jiliang University, Hangzhou, 310018, P.R. China

2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. China

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Journal of Inequalities and Applications 2012, 2012:145 doi:10.1186/1029-242X-2012-145

 Received: 12 January 2012 Accepted: 6 June 2012 Published: 22 June 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In the present paper we establish new inequalities similar to the extensions of Hilbert’s double-series inequality and also give their integral analogues. Our results provide some new estimates to these types of inequalities.

MSC: 26D15.

##### Keywords:
Hilbert’s inequality; Pachpatte’s inequality; Hölder’s inequality

### 1 Introduction

In recent years several authors have given considerable attention to Hilbert’s double-series inequality together with its integral version, inverse version, and various generalizations (see [1-9]). In this paper, we establish multivariable sum inequalities for the extensions of Hilbert’s inequality and also obtain their integral forms. Our results provide some new estimates to these types of inequalities.

The well-known classical extension of Hilbert’s double-series theorem can be stated as follows [10], p.253].

Theorem AIfare real numbers such thatand, where, as usual, andare the conjugate exponents ofandrespectively, then

(1.1)

wheredepends onandonly.

In 2000, Pachpatte [11] established a new inequality similar to inequality (1.1) as follows:

Theorem A′Letpqandbe as in[11], then

(1.2)

The integral analogue of inequality (1.1) is as follows [10], p.254].

Theorem BLetp, q, , andλbe as in Theorem A. Ifand, then

(1.3)

wheredepends onpandqonly.

In [11], Pachpatte also established a similar version of inequality (1.3) as follows.

Theorem B′Letpqandbe as in[11], then

(1.4)

In the present paper we establish some new inequalities similar to Theorems A, A, B and B. Our results provide some new estimates to these types of inequalities.

### 2 Statement of results

Our main results are given in the following theorems.

Theorem 2.1Letbe constants and. Letbe real-valued functions defined for, where () are natural numbers. For convenience, we writeand. Define the operatorsbyfor any function. Then

(2.1)
where

Remark 2.1 Let change to in Theorem 2.1 and in view of and for any function , , then

(2.2)

where

Remark 2.2 Taking for in Remark 2.1. If satisfy and , then inequality (2.2) reduces to

(2.3)

which is an interesting variation of inequality (1.1).

On the other hand, if , then and so , . In this case inequality (2.3) reduces to

This is just a similar version of inequality (1.2) in Theorem A.

Theorem 2.2Letbe constants and. Letbe real-valuednth differentiable functions defined on, where, and. Suppose

then

(2.4)
where

Remark 2.3 Let change to in Theorem 2.2 and in view of , , then

(2.5)

where

Remark 2.4 Taking for in Remark 2.3, if are such that and , inequality (2.5) reduces to

(2.6)

which is an interesting variation of inequality (1.3).

On the other hand, if , then and so , . In this case inequality (2.6) reduces to

This is just a similar version of inequality (1.4) in Theorem B.

### 3 Proofs of results

Proof of Theorem 2.1 From the hypotheses , we have

(3.1)

From the hypotheses of Theorem 2.1 and in view of Hölder’s inequality (see [10]) and inequality for mean [10], we obtain

(3.2)

Dividing both sides of (3.2) by and then taking sums over from 1 to (), respectively and then using again Hölder’s inequality, we obtain

This concludes the proof. □

Proof of Theorem 2.2 From the hypotheses of Theorem 2.2, we have

(3.3)

On the other hand, by using Hölder’s integral inequality (see [10]) and the following inequality for mean [10],

we obtain

(3.4)

Dividing both sides of (3.4) by and then integrating the result inequality over from 1 to (), respectively and then using again Hölder’s integral inequality, we obtain

This concludes the proof. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

C-JZ and W-SC jointly contributed to the main results Theorems 2.1 and 2.2. All authors read and approved the final manuscript.

### Acknowledgement

CJZ is supported by National Natural Science Foundation of China (10971205). WSC is partially supported by a HKU URG grant. The authors express their grateful thanks to the referees for their many very valuable suggestions and comments.

### References

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