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A multiple Hilbert-type integral inequality with a non-homogeneous kernel
Journal of Inequalities and Applications volume 2013, Article number: 73 (2013)
By using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral inequality with a non-homogeneous kernel is given. The operator expression with the norm, the reverses and some examples with the particular kernels are also considered.
If , , , , , then we have the following equivalent inequalities (cf. ):
where the constant factor is the best possible. (1) is the well-known Hardy-Hilbert integral inequality. Define the Hardy-Hilbert integral operator as follows: for , ().
Then in view of (2), it follows and . Since the constant is the best possible, we find .
Inequalities (1) and (2) and the operator are important in analysis and its applications (cf. [2, 3]). In 2002,  considered the property of the Hardy-Hilbert integral operator and gave an improvement of (1) (for ). In 2004, by introducing another pair of conjugate exponents (, ) and an independent parameter ,  gave the best extensions of (1) as follows:
where , , , . In 2007,  gave the following inequality with the best constant (; is the beta function):
In 2009,  gave an extension of (4) in with the kernel ();  gave another extension of (4) to the general kernel () with a pair of conjugate exponents and obtained the following multiple Hilbert-type integral inequality. Suppose that , , , , is a measurable function of −λ-degree in , and for any (), satisfies and
If , , (), then we have the following inequality:
where the constant factor is the best possible. For , in (5), we obtain (3). Inequality (5) is an extension of the results in [9–12] and . In recent years,  and  considered some Hilbert-type operators relating (1)-(3);  also considered a multiple Hilbert-type integral operator with the homogeneous kernel of -degree and the relating particular case of (5) (for , ).
In this paper, by using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral inequality with a non-homogeneous kernel is given. The operator expression with the norm, the reverses and some examples with the particular kernels are considered.
2 Some lemmas
Lemma 1 If , (), , then we have
and then (6) is valid. □
Definition 1 If , , , is a measurable function in such that for any and , , then we call the homogeneous function of −λ-degree in .
Lemma 2 Suppose , (), , is a homogeneous function of −λ-degree. If
satisfying , then each and for any ,
Proof Setting () in the integral , we find
Setting in the above integral, we obtain . Setting in (7), since , we find
Setting () and in the above integral, we find . □
Lemma 3 With the assumptions given in Lemma 2, then
is finite in a neighborhood of if any only if is continuous at .
Proof The sufficiency property is obvious. We prove the necessary property of the condition by mathematical induction in the following. For , there exists such that for any , . Since for (),
and , then by the Lebesgue control convergence theorem (cf. ), it follows (). Assuming that for n (≥2), is continuous at , then for , by the result of , since is finite in a neighborhood of , we find
then by the assumption for n, it follows
By mathematical induction, we prove that for , is continuous at . □
Lemma 4 With the assumptions given in Lemma 2, if there exists such that for , , (), , then we have
Proof Setting in (8), we find
Setting () in the above integral, since , we find (replacing by )
by (9), it follows
Without loss of generality, we estimate the case of , e.t.
. In fact, setting , such that , since (), there exists such that (), and then by the Fubini theorem, it follows
Hence by (10), we have
Since by Lemma 3 we find
then combining with (11), we have (8). □
Lemma 5 Suppose that , (), , , , , (≥0) is a measurable function of −λ-degree in such that
If are measurable functions in (), , then (1) for (), we have
(2) for , (), we have the reverse of (12).
Proof (1) For (), by the Hölder inequality (cf. ) and (7), it follows
For , by the Hölder inequality again, it follows
Then by (7), we have (12). (Note: for , we do not use the Hölder inequality again in the above.) (2) For , (), by the reverse Hölder inequality and in the same way, we obtain the reverses of (12). □
3 Main results and applications
With the assumptions given in Lemma 5, setting (; ), then we find . If (), define the following real function spaces:
and a multiple Hilbert-type integral operator as follows: for ,
Then by (12), it follows , T is bounded, and , where
Define the formal inner product of and as
Theorem 1 With the assumptions given in Lemma 5, suppose that for any , it satisfies , and
If , (), then (i) for (), we have and the following equivalent inequalities:
where the constant factor is the best possible; (ii) for , (), using the formal symbols in the case of (i), we have the equivalent reverses of (19) and (20) with the same best constant factor.
Proof (i) For all , if (12) takes the form of equality, then for in (14), there exist and () such that they are not all zero and
e.t. a.e. in . Assuming that , then , which contradicts . (Note: for , we consider (13) for in the above.) Hence we have (19). By the Hölder inequality, it follows
and then by (19), we have (20). Assuming that (20) is valid, setting
then . By (12), it follows . If , then (19) is naturally valid. Assuming that , by (20), it follows
and then (19) is valid, which is equivalent to (20).
For small enough, setting as: , ; , (), , ; , , if there exists such that (20) is still valid as we replace by k, then in particular, by Lemma 4, we have
and . Hence is the best value of (20). We confirm that the constant factor in (19) is the best possible, otherwise we can get a contradiction by (21) that the constant factor in (20) is not the best possible. Therefore .
For , (), by using the reverse Hölder inequality and in the same way, we have the equivalent reverses of (19) and (20) with the same best constant factor. □
Example 1 For , (), , , , by mathematical induction, we can show
In fact, for , we obtain
Assuming that for n (≥2) (22) is valid, then for , it follows
Then by mathematical induction, (22) is valid for .
Example 2 For , (), , , , by mathematical induction, we can show
In fact, for , we obtain
Assuming that for n (≥2), (22) is valid, then for , it follows
Then by mathematical induction, (23) is valid for .
Example 3 For , (), , , , by mathematical induction, we can show
In fact, for , we obtain
Assuming that for n (≥2), (24) is valid, then for , it follows
Then, by mathematical induction, (24) is valid for .
Remarks (i) In particular, for in (20), we have
where () is the best possible. Inequality (25) is an extension of (4) and (8.1.7) in .
In Examples 1 and 2, by Theorem 1, since for any , we obtain , then we have and the equivalent inequalities (19) and (20) with the particular kernels and some equivalent reverses. In Example 3, still using Theorem 1, we find and the relating particular inequalities.
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This work is supported by Guangdong Modern Information Service industry Develop Particularly item 2011 (No. 13090).
The authors declare that they have no competing interests.
QH carried out the study, and wrote the manuscript. BY participated in the design of the study, and reformed the manuscript. All authors read and approved the final manuscript.
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Huang, Q., Yang, B. A multiple Hilbert-type integral inequality with a non-homogeneous kernel. J Inequal Appl 2013, 73 (2013). https://doi.org/10.1186/1029-242X-2013-73
- multiple Hilbert-type integral inequality
- weight function