A Hilbert-Type Integral Inequality with Multiparameters and a Nonhomogeneous Kernel

Abstract

We first introduce -function and Riemann -function to characterize the constant factor jointly. A Hilbert-type integral inequality with multiparameters and a nonhomogeneous kernel is given using the way of weight function and the technique of real analysis. The equivalent form is considered and its constant factors are proved to be the best possible. Some meaningful results are obtained by taking the special parameter values.

1. Introduction

If is a measurable function and is a positive number, set the function spaces as follows:

If , , , then we have [1] where the constant factor is the best possible. Inequality (2) is the famous Hilbert integral inequality, which is important in analysis and its applications [1, 2]. We note that the kernel of (2) is homogeneous of −1-degree, and the methods and results about Hilbert-type inequality with the negative homogeneous kernel and multiparameters are summarized systematically in the paper in [3]. In recent years, Hilbert-type inequality was researched from homogeneous kernel turn to nonhomogeneous kernel [4–10], such that two integral inequalities with the best constant factor are obtained in [6, 7] as follows:

If , , , , , , then

In this paper, a Hilbert-type integral inequality with nonhomogeneous kernel is given using the way of weight function and the technique of real analysis as follows.

If , , , , , , then

2. Some Lemmas

Lemma 1. Let , ; then, the Laplace integral transform of the power function is as follows [11]: where is the -function .

Lemma 2. Let be a positive integer; then, one has the summation formula [12] and [8] where the are the Bernoulli numbers, namely, , , , , , , , and so forth.

Lemma 3. If , , , define the weight function as follows: then, one has where particularly.
(1) When , by (11), we find that where is Riemann -function .
(2) When , , from (8) and (12), we find that where the are the Bernoulli numbers, namely, , , , , , , , and so forth.

Proof. Setting , then, by (6), we have By the same way, we obtain .

Lemma 4. If , , , , and is small enough, define the real functions as follows: then, one finds that

Proof. We easily get Since is continuous on and , , there exists , such that . From Fubini theorem [13], we have

3. Main Results and Applications

Theorem 5. If , , , , , , , , then one has where the constant factor (with (11)) is the best possible.

Proof. By Hölder’s inequality [14], Fubini theorem, and Lemma 4, we obtain
If inequality (21) keeps the form of equality, then there exist constants and [14], which are not all zeroes, such that a.e. in . It follows that a.e. on . Assuming that , there exists , such that a.e. in , which contradicts the fact that . Then, inequality (21) keeps the strict form.
If the constant factor of (20) is not the best possible, then there exists a positive, such that inequality (20) is still valid if we replace by ; then, by (16) and (17), we have Letting , we get , which contradicts the fact that , so the constant factor of (20) is the best possible.

Theorem 6. If , , , , , then one has where the constant factor is the best possible, and inequality (23) is equivalent to inequality (20).

Proof. Setting a bounded measurable function as since , there exists , such that . Setting when , from (20), we find that It follows that . For , by (20), (26) still keep the forms of strict inequality. Hence, we have (23).
On the other hand, from Hölder inequality, we find that
The inequality is (20), which is equivalent to (23).
If the constant factor of (23) is not the best, by (23), then we obtain that the constant factor of (20) is not the best too, which contradicts Theorem 5. Thus, the constant factor in (23) is the best possible.

By taking the special parameter values in (20) and (23), some meaningful inequalities are obtained as follows.

Example 7. Let , ; by (8) and (11), we get . If , , , , , then we have (5) and its equivalent form as where the constant factor is the best possible.

Example 8. Let , , ; by (12), we get . If , , , , , , then we have the following equivalent inequalities: where the constant factors , in (29) are the best possible.

Example 9. Let , , ; by (13), we get . If , , , , , , then we have the following equivalent inequalities: where the constant factors , in (30) are the best possible.

Conflict of Interests

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

Acknowledgment

This paper was funded by the National Natural Science Foundation of China (no. 11171280).