On a half-discrete Hilbert-type inequality similar to Mulholland’s inequality
© Huang and Yang; licensee Springer 2013
Received: 23 January 2013
Accepted: 30 April 2013
Published: 7 June 2013
By using the way of weight functions and Hadamard’s inequality, a half-discrete Hilbert-type inequality similar to Mulholland’s inequality with a best constant factor is given. The extension with multi-parameters, the equivalent forms as well as the operator expressions are also considered.
Moreover, the best extension of (7) with multi-parameters, some equivalent forms as well as the operator expressions are considered.
2 Some lemmas
namely, (10) follows. □
and then in view of (10), inequality (12) follows. □
3 Main results
wherefrom, , and .
where the constant is the best possible in the above inequalities.
that is, (15) is equivalent to (13). Hence, inequalities (13), (14) and (15) are equivalent.
and (). Hence is the best value of (13).
By equivalence, the constant factor in (14) and (15) is the best possible. Otherwise, we can imply a contradiction by (16) and (17) that the constant factor in (13) is not the best possible. □
- (ii)Define the second type half-discrete Hilbert-type operator as follows: For , we define , satisfying
Then by (15) it follows , and then is a bounded operator with . Since by Theorem 1 the constant factor in (15) is the best possible, we have .
This work is supported by 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).
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