- Open Access
A multiple Hilbert-type integral inequality with a non-homogeneous kernel
© Huang and Yang; licensee Springer 2013
- Received: 3 October 2012
- Accepted: 11 February 2013
- Published: 28 February 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.
- multiple Hilbert-type integral inequality
- weight function
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 .
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.
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 .
Setting () and in the above integral, we find . □
is finite in a neighborhood of if any only if is continuous at .
By mathematical induction, we prove that for , is continuous at . □
Without loss of generality, we estimate the case of , e.t.
then combining with (11), we have (8). □
(2) for , (), we have the reverse of (12).
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). □
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.
and then (19) is valid, which is equivalent to (20).
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. □
Then by mathematical induction, (22) is valid for .
Then by mathematical induction, (23) is valid for .
Then, by mathematical induction, (24) is valid for .
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.
This work is supported by Guangdong Modern Information Service industry Develop Particularly item 2011 (No. 13090).
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