Half-discrete Hardy-Hilbert’s inequality with two interval variables
© Chen and Yang; licensee Springer. 2013
Received: 2 July 2013
Accepted: 17 September 2013
Published: 7 November 2013
By using the way of weight functions and the technique of real analysis, a half-discrete Hardy-Hilbert’s inequality with two interval variables is derived. The equivalent forms, operator expressions, some reverses as well as a few particular cases are obtained.
Regarding the case of half-discrete Hilbert-type inequalities with non-homogeneous kernels, Hardy, Littlewood and Polya provided some results in Theorem 351 of . However, they had not proved that the constant factors in the new inequalities were best possible. Yang  proved some results by introducing an interval variable and that the constant factors are best possible.
The best extension of (5) with two interval variables, some equivalent forms, operator expressions, some reverses as well as a few particular cases are also considered.
2 Some lemmas
Hence, we have (8) and (9). □
- (i)For , we have the following inequalities:(10)(11)
For , we have the reverses of (10) and (11).
and (10) follows.
By reverse Hölder’s inequality (cf. ) and in the same way, for , we can obtain the reverses of (10) and (11). □
3 Main results
where the constant factor is best possible.
Proof By the Lebesgue term-by-term integration theorem, there are two expressions for I in (12). In view of (8) and (10), we obtain (13).
and thus we get (13), which is equivalent to (12).
In view of (8) and (11), we have (14).
and we have (14), which is equivalent to (12).
Hence inequalities (12), (13) and (14) are equivalent.
and then (). Hence is the best possible constant factor of (12).
We conform that the constant factor in (13) ((14)) is best possible. Otherwise, we would reach a contradiction by (15) ((16)) that the constant factor in (12) is not best possible. □
- (i)Define a half-discrete Hilbert’s operator as follows: , for , there exists a unified representation satisfying
- (ii)Define a half-discrete Hilbert’s operator as follows:
Since the constant factor in (14) is best possible, we have .
In the following theorem, for , we still use the formal symbols of and et al.
Moreover, if there exists a constant such that for any , is decreasing in , then the constant factor in the above inequalities is best possible.
we have (20).
and we have (20), which is equivalent to (19).
we have (21).
and we have (21), which is equivalent to (19).
Hence inequalities (19), (20) and (21) are equivalent.
Hence is the best possible constant factor of (19).
We conform that the constant factor in (20) ((21)) is best possible. Otherwise, we would reach a contradiction by (22) ((23)) that the constant factor in (19) is not best possible. □
- (ii)For , , , , , in (12), (13) and (14), we have the following half-discrete Mulholland’s inequality and its equivalent forms:(28)
- (iii)For , in (5), (26) and (27), we can obtain the following equivalent inequalities with non-homogeneous kernel and the best constant factor :(31)
In fact, we can show that (31), (32) and (33) are respectively equivalent to (5), (26) and (27), and then it follows that (31), (32) and (33) are equivalent with the same best constant factor .
This work is supported by the National Natural Science Foundation of China (No. 61370186), 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079), Science and Technology Application Foundation Program of Guangzhou (No. 2013J4100009) and the Ministry of Education and China Mobile Research Fund (No. MCM20121051).
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