On a half-discrete reverse Mulholland-type inequality and an extension
© Liu et al.; licensee Springer. 2014
Received: 19 November 2013
Accepted: 6 February 2014
Published: 4 March 2014
By using the way of weight functions and the Hermite-Hadamard inequality, a half-discrete reverse Mulholland-type inequality with a best constant factor is given. The extension with multi-parameters, the equivalent forms as well as the relating homogeneous inequalities are also considered.
KeywordsMulholland-type inequality weight function equivalent form reverse
with the same best constant factor . Clearly, for , , , , (4) reduces to (1), while (5) reduces to (2).
Moreover, a best extension of (7) with multi-parameters, the equivalent forms and the relating homogeneous inequalities are considered.
2 Some lemmas
that is, (10) is valid. □
and then in view of (10), inequality (12) follows. □
3 Main results
wherefrom and .
where the constant is best possible.
that is, (15) is equivalent to (13). Hence, inequalities (13), (14) and (15) are equivalent.
and . Hence is the best value of (13).
By the equivalence, the constant factor in (14) and (15) is best possible. Otherwise we would reach a contradiction by (16) and (17) that the constant factor in (13) is not best possible. □
This work is supported by the National Natural Science Foundation of China (No. 61370186).
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