On a more accurate half-discrete mulholland's inequality and an extension
© Chen and Yang; licensee Springer. 2012
Received: 14 November 2011
Accepted: 26 March 2012
Published: 26 March 2012
By using the way of weight functions and Jensen-Hadamard's inequality, a more accurate half-discrete 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 considered.
Mathematics Subject Classication 2000: 26D15; 47A07.
Moreover, a best extension of (7) with multi-parameters, some equivalent forms as well as the operator expressions are also considered.
2 Some lemmas
namely, (10) follows. □ ▪
and then in view of (10), inequality (12) follows. □ ▪
3 Main results
wherefrom, , and .
where the constant B(λ1, λ2) is the best possible in the above inequalities.
That is, (15) is equivalent to (13). Hence inequalities (13), (14) and (15) are equivalent.
and B(λ1, λ2) ≤ k(ε → 0+). Hence, k = B(λ1, λ2) is the best value of (13).
Due to the equivalence, the constant factor B(λ1, λ2) 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. □ ▪
Then by (14), it follows and then T is a bounded operator with ║T║ ≤ B(λ1, λ2). Since by Theorem 1, the constant factor in (14) is the best possible, we have ║T║ = B(λ1, λ2).
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 the Guangdong Science and Technology Plan Item (No. 2010B010600018), and the Guangdong Modern Information Service industry Develop Particularly item 2011 (No. 13090).
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