On a Multiple Hilbert-Type Integral Operator and Applications
© Q. Huang and B. Yang. 2010
Received: 19 August 2009
Accepted: 9 December 2009
Published: 12 January 2010
By using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral operator with the homogeneous kernel of -degree ( ) and its norm are considered. As for applications, two equivalent inequalities with the best constant factors, the reverses, and some particular norms are obtained.
where the constant factor is the best possible. Define the Hardy-Hilbert's integral operator as follows: for Then in view of (1.2), it follows that and Since the constant factor in (1.2) is the best possible, we find that (cf.)
where the constant factor is the best possible. For , and in (1.6), we obtain (1.3) and (1.4). Inequality (1.6) is some extensions of the results in [6, 8–11]. In 2006, reference  also considered a multiple Hilbert-type integral operator with the homogeneous kernel of -degree and its inequality with the norm, which is the best extension of (1.2).
In this paper, by using the way of weight functions and the technic of real analysis, a new multiple Hilbert-type integral operator with the norm is considered, which is an extension of the result in . As for applications, an extended multiple Hilbert-type integral inequality and the equivalent form, the reverses, and some particular norms are obtained.
2. Some Lemmas
and then (2.1) is valid.
Then by combination with (2.15), we have (2.10).
3. Main Results and Applications
and then (3.6) is valid, which is equivalent to (3.7).
and Hence is the best value of (3.7). We conform that the constant factor in (3.6) is the best possible; otherwise, we can get a contradiction by (3.9) that the constant factor in (3.7) is not the best possible. Therefore
This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026), and Guangdong Natural Science Foundation (no. 7004344).
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