On a New Hilbert-Hardy-Type Integral Operator and Applications
© Xingdong Liu and Bicheng Yang. 2010
Received: 7 September 2010
Accepted: 26 October 2010
Published: 27 October 2010
By applying the way of weight functions and a Hardy's integral inequality, a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-Hardy-type inequality similar to Hilbert-type integral inequality is given, and two equivalent inequalities with the best constant factors as well as some particular examples are considered.
In 1934, Hardy published the following theorem (cf. [1, Theorem 319]).
Hardy  also published the following Hardy's integral inequality.
where the constant factor is the best possible (cf. [1, Theorem 330]).
In 2009, Yang  published the following theorem.
where the constant factor is the best possible (cf. ). Sulaiman  also considered a Hilbert-Hardy-type integral inequality similar to (1.4) with the kernel , , . But he cannot show that the constant factor in the new inequality is the best possible.
In this paper, by applying the way of weight functions and inequality (1.2) for , a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-Hardy-type inequality similar to (1.3) is given, and two equivalent inequalities with a best constant factor as well as some particular examples are considered.
2. A Lemma and Two Equivalent Inequalities
The lemma is proved.
Then by (2.7), we have (2.6).
Hence, we have (2.7), which is equivalent to (2.6).
3. A Hilbert-Hardy-Type Integral Operator and Applications
Then, by (2.6), we have (3.4).
We conclude that the constant factor in (2.6) is the best possible, otherwise we can get a contradiction by (1.2) that the constant factor in (3.4) is not the best possible. By the same way, if the constant factor in (2.7) is not the best possible, then by (2.13), we can get a contradiction that the constant factor in (2.6) is not the best possible. Therefore in view of (3.3), we have (3.5).
where the constant factors in the above inequalities are the best possible.
This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026) and the Guangdong Natural Science Foundation (no. 7004344).
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