- Open Access
On a more accurate half-discrete Hilbert's inequality
© Huang and Yang; licensee Springer. 2012
- Received: 8 March 2012
- Accepted: 8 May 2012
- Published: 8 May 2012
By using the way of weight coefficients and the idea of introducing parameters and by means of Hadamard's inequality, we give a more accurate half-discrete Hilbert's inequality with a best constant factor. We also consider its best extension with parameters, the equivalent forms, the operator expressions as well as some reverses.
2000 Mathematics Subject Classification: 26D15; 47A07.
- weight coefficient
- equivalent form
- Hilbert's inequality
- Hadamard's inequality
where the constant factor π is still the best possible. Inequalities (1)-(3) are important in analysis and its applications . There are lots of improvements, generalizations, and applications of inequalities (1-3), for more details, refer to literatures [5–18].
where the constant factor π is the best possible.
We also consider its best extension with parameters, the equivalent forms, the operator expressions as well as some reverses.
Hence, we prove that (9) is valid.
where ϖ(x) and ω(n) are indicated by (7) and (8).
(ii) for p < 1(p ≠ 0), we have the reverses of (13) and (14).
Hence (14) is valid.
(ii) For 0 < p < 1(q < 0) or p < 0(0 < q < 1), using the reverse Hölder's inequality and in the same way, we have the reverses of (13) and (14).
Then by (26) and (27), (21) is valid.
In virtue of (30) and (31), (23) is valid.
where the constant factor is the best possible.
Hence (33) is valid, which is equivalent to (32).
Hence (34) is valid, which is equivalent to (32). It follows that (32), (33), and (34) are equivalent.
If there exists a positive number , such that (32) is still valid as we replace , by k, then in particular, (20) is valid ( are taken as (19)). Then we have (21). For ε → 0+ in (21), we have Hence, is the best value of (32). We conform that the constant factor in (33) [(34)] is the best possible, otherwise we can get a contradiction by (35) [(39)] that the constant factor in (32) is not the best possible.
- (ii)Define a half-discrete Hilbert's operator in the following way: For a ∈ l q , ψ , we define , satisfying
Then by (34), it follows i.e. is the bounded operator with Since the constant factor in (34) is the best possible, we have
where the constant factor is the best possible.
Then by (43), (42) is valid.
Hence (43) is valid, which is equivalent to (42).
Hence (44) is valid, which is equivalent to (42). It follows that (42), (43), and (44) are equivalent.
If there exists a positive number such that (42) is still valid as we replace by k, then in particular, (22) is valid. Hence we have (23). For ε → 0+ in (23), we obtain Hence is the best value of (42). We conform that the constant factor in (43) [(44)] is the best possible, otherwise we can get a contradiction by (45) [(46)] that the constant factor in (42) is not the best possible.
In the same way, for p < 0, we also have the following result:
Theorem 3 If the assumption of p > 1 in Theorem 1 is replaced by p < 0, then the reverses of (32), (33), and (34) are valid and equivalent. Moreover, the same constant factor is the best possible.
In particular, for , , p = q = 2 in (48), we obtain (5). Hence, inequality (32) is the best extension of (4) and (5) with parameters.
This study was 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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