Abstract and Applied Analysis

Volume 2013 (2013), Article ID 861948, 6 pages

http://dx.doi.org/10.1155/2013/861948

Research Article

Inequalities Similar to Hilbert's Inequality

Chang-Jian Zhao

Department of Mathematics, China Jiliang University, Hangzhou 310018, China

Received 23 June 2013; Accepted 4 August 2013

Academic Editor: Wenchang Sun

Copyright © 2013 Chang-Jian Zhao. 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.

Abstract.

In the present paper, we establish some new inequalities similar to Hilbert’s type inequalities. Our results provide some new estimates to these types of inequalities.

1. Introduction

The well-known classical Hilbert’s double-series inequality can be stated as follows [1, page 253].

Theorem A. If such that and , where, as usual, and are the conjugate exponents of and , respectively,  then

where depends on and only.

In recent years, several authors [1–18] have given considerable attention to Hilbert’s double-series inequality together with its integral version, inverse version, and various generalizations. In particular, Pachpatte [11] established an inequality similar to inequality (1) as follows.

Theorem 1. Let be constant and . If and are real-valued functions defined for and , respectively, and . Moreover, define the operators by . Then,

The first aim of this paper is to establish a new inequality similar to Hilbert’s type inequality. Our result provides new estimates to this type of inequality.

Theorem 2. Let be constants, and . For , let be real-valued functions defined for , where ;  , and let , be natural numbers. Let , and define the operators by

Then,

where

Remark 3. Inequality (4) is just a similar version of the following inequality established by Pachpatte [11]:

On the other hand, let and change to and , respectively, and, with appropriate transformation, we have

where

and is as in (6). This is just a similar version of inequality (2) in Theorem 1.

The integral analogue of inequality (1) in Theorem A is as follows [1, page 254].

Theorem B. Let , , , , and be as in Theorem A. If and , then

where depends on and only.

In [11], Pachpatte also established a similar version of inequality (10) as follows.

Theorem 4. Let be constants, and . If and are real-valued continuous functions defined on and , respectively, and let . Then,

Another aim of this paper is to establish a new integral inequality similar to Hilbert’s type inequality.

Theorem 5. Let , and . For , let ,   be real-valued differentiable functions defined on , where , and . As usual, partial derivatives of are denoted by , and so forth. Let

Then,

where

and is as in (6).

Remark 6. Inequality (13) is just a similar version of the following inequality established by Pachpatte [11]:

On the other hand, let and change to and , respectively, and, with appropriate transformation, we have

where

This is just a similar version of inequality (11) in Theorem 4.

2. Proof of Theorems

Proof of Theorem 2. From the hypotheses of Theorem 2, we have

By using Hölder’s inequality and noticing the reverse Young’s inequality [19],

for positive real numbers , and , , where is as in (6). Hence,

Dividing both sides of (20) by

taking the sum of both sides of (20) over and from 1 to and , respectively, and making use of Hölder’s inequality, we have

This completes the proof.

Proof of Theorem 5. From the hypotheses of  Theorem 5, we obtain for :

From (23), Hölder’s integral inequality and in view of the reverse Young’s inequality (19), we have

Integrating both sides of (24) over and from to and (), respectively, and by using Hölder’s integral inequality, we arrive at

This completes the proof.

Acknowledgment

The research is supported by the National Natural Science Foundation of China (11371334).

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