Research Article | Open Access

# On Hardy-Pachpatte-Copson's Inequalities

**Academic Editor:**D. Xu

#### Abstract

We establish new inequalities similar to Hardy-Pachpatte-Copson’s type inequalities. These results in special cases yield some of the recent results.

#### 1. Introduction

The classical Hardy’s integral inequality is as follows.

Theorem A. *If , for , and , then
**
unless . The constant is the best possible.*

Theorem A was first proved by Hardy [1], in an attempt to give a simple proof of Hilbert's double series theorem (see [2]). One of the best known and interesting generalization of the inequality (1) given by Hardy [3] himself can be stated as follows.

Theorem B. *If , for , and is defined by
**
then
**
unless . The constant is the best possible.*

Inequalities (1) and (3) which later went by the name of Hardy’s inequalities led to a great many papers dealing with alternative proofs, various generalizations, and numerous variants and applications in analysis (see [4–15]). In particular, Pachpatte [4] established some generalizations of Hardy inequalities (1) and (3). Very recently, Leng and Feng [16] proved some new Hardy-type integral inequalities. In the present paper we establish new inequalities similar to Hardy's integral inequalities (1) and (3). These results provide some new estimates to these types of inequalities and in special cases yield some of the recent results.

#### 2. Main Results

Our main results are given in the following theorems.

Theorem 1. *Let , , , , and be constants. Let be positive and locally absolutely continuous in . Let be a positive continuous function and let , for . Let be nonnegative and measurable on . If
**
for almost all , and if is defined by
**
for , then
**
where
*

*Remark 2. *Let , , , and reduce to , , , and , respectively, and with suitable modifications in Theorem 1, (6) changes to the following result:

This is just a new inequality established by Pachpatte [4].

Moreover, we note that the inequality established in Theorem 1 is the further generalizations of the inequality established by Copson [17].

Taking for , , and in (8), (8) changes to the following result:

This is just a new inequality established by Love [7].

Let , , , and in (9); then (9) changes to the following result:

This result is obtained in (3) stated in the Introduction.

Theorem 3. *Let , , , , and be constants. Let be positive and locally absolutely continuous in . Let be a positive continuous function and let , for . Let be nonnegative and measurable on . Let
**
for almost all . If is defined by
**
for , then
**
where
*

*Remark 4. *Let , and reduce to , , , and , respectively, and with suitable modifications in Theorem 3, (13) changes to the following result:

This is just a new inequality established by Pachpatte [4].

On the other hand, we note that the inequality established in Theorem 3 is the further generalizations of the inequality established by Copson [17].

Taking for , and in (15), (15) changes to the following result:

This is just a new inequality established by Love [7].

#### 3. Proof of Theorems

* Proof of Theorem 1. *If we let and in view of
for , then

Let
where and in view of , for , then

From (18), (20), and integrating by parts for , we have
where

If , then we observe that

By applying Hölder’s inequality with indices , on the right side of (23), we obtain

Dividing both sides of (24) by the second integral factor on the right side of (24) and raising both sides to the th power, we obtain

*Proof of Theorem 3. *If we let and in view of
for , then

Let
where and in view of , for , then

From (27), (29), and integrating by parts for , we have
where

If , then we observe that

By applying Hölder’s inequality with indices , on the right side of (32), we obtain

Dividing both sides of (33) by the second integral factor on the right side of (33) and raising both sides to the th power, we obtain

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

Chang-Jian Zhao’s research is supported by National Natural Science Foundation of China (11371334). Wing-Sum Cheung’s research is partially supported by a HKU URG grant.

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#### Copyright

Copyright © 2014 Chang-Jian Zhao and Wing-Sum Cheung. 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.