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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 381650, 8 pages
http://dx.doi.org/10.1155/2008/381650
Research Article

On a Generalization of Hilbert-Type Integral Inequality

Basic Courses Department, Zhejiang Water Conservancy & Hydropower College, Hangzhou 310018, China

Received 1 March 2008; Accepted 9 May 2008

Academic Editor: Feng Qi

Copyright © 2008 Sun Baoju. 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

By introducing some parameters, we establish generalizations of the Hilbert-type inequality. As applications, the reverse and its equivalent form are considered.

1. Introduction

Considerable attention has been given to Hilbert inequalities and Hilbert-type inequalities by several authors including Gao and Yang [1], Yang [24], Jichang and Debnath [5], Pachpatte [6], Zhao [7], Brnetić and Pečarić [8]. In 2007, Li et al. [9] gave a new inequality similar to Hilbert inequality for integrals:

If , then one has

where constant factor 4 is the best possible.

An equivalent inequality is

In this paper, by introducing some parameters we generalize (1.1), (1.2), and we obtain the reverse form for each of them. The equivalent forms are also considered.

2. Main Results

Lemma 2.1. Suppose that , define weight functions , , respectively as
One has

Proof. Letting , we have
By symmetry we have
The lemma is proved.

Lemma 2.2. Let , , and setting
Then for one gets

Proof. Letting , we have
Now, observe that
then
We get
On the other hand,
Hence, (2.5) is valid. The lemma is proved.

Theorem 2.3. Let , , , If , , then one has
where constant factor is the best possible. In particular, for inequality (2.11) reduces to

Proof. Applying Hölder’s inequality and Lemma 2.1, we have

If (2.13) takes the form of equality, then there exist constants and , such that they are not all zero, and
a.e. in . It follows that there exists a constant , such that
Without lose of generality, suppose , then we have
which contradicts the fact that hence (2.13) takes the form of strict inequality, so we obtain (2.11).
Assume that the constant factor in (2.11) is not the best possible, then there exists a positive number (with ) such that (2.11) is still valid if one replaces by . In particular, for , setting and as for , , for then we have
By using Lemma 2.2, we find
Therefore, we get
or
For it follows that This contradicts the fact that Hence, the constant factor in (2.11) is the best possible. Theorem 2.3 is proved.

Theorem 2.4. Let , , , If , , then one has
where the constant factor is the best possible. In particular, for the inequality reduces to

Proof. Applying reverse Hölder’s inequality and the same arguments as before, we have (2.21).
If the constant factor in (2.21) is not the best possible, then there exists a positive number (with ), such that (2.21) is still valid if one replaces by . In particular, for , setting and as in Theorem 2.3, we have
By using Lemma 2.2, we find
Therefore, we get
for and it follows that This contradicts the fact that Hence, the constant factor in (2.21) is the best possible. Theorem 2.4 is proved.

Theorem 2.5. If , , , , then one has
where the constant factor is the best possible. Inequality (2.26) is equivalent to (2.11).

Proof. Setting
then by (2.11), we find
Hence, we obtain
Thus, by (2.11), both (2.28) and (2.29) keep the form of strict inequalities, then we have (2.26).
Applying Hölder’s inequality, we have
Therefore, by (2.26) we have (2.11). It follows that inequality (2.26) is equivalent to (2.11), and the constant factors in (2.26) are the best possible. The theorem is proved.

Theorem 2.6. If , , , , then one has
where the constant factor is the best possible. Inequality (2.31) is equivalent to (2.21).

The proof of Theorem 2.6 is similar to that of Theorem 2.5, so we omit it.

Acknowledgment

The author would like to thank the anonymous referees for their suggestions and corrections.

References

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