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On a half-discrete reverse Mulholland-type inequality and an extension
Journal of Inequalities and Applications volume 2014, Article number: 103 (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.
Assuming that , , , we have the following Hilbert integral inequality (cf. ):
where the constant factor π is best possible. If , , , then we still have the following discrete Hilbert inequality:
with the same best constant factor π. Inequalities (1) and (2) are important in analysis and its applications (cf. [2–4]). Also we have the following Mulholland inequality with the same best constant factor π (cf. [1, 5]):
In 1998, by introducing an independent parameter , Yang  gave an extension of (1). For generalizing the results from , Yang  gave some best extensions of (1) and (2) as follows. If , , , is a non-negative homogeneous function of degree −λ, , , , , , , then
where the constant factor is best possible. Moreover, if is finite and is strict decreasing for (), then for , , , , we have
with the same best constant factor . Clearly, for , , , , (4) reduces to (1), while (5) reduces to (2).
Some other results including the reverse Hilbert-type inequalities are provided by [8–16]. On half-discrete Hilbert-type inequalities with the non-homogeneous kernels, Hardy et al. provided a few results in Theorem 351 of . But they did not prove that the constant factors in the inequalities are best possible. However, Yang  gave a result by introducing an interval variable and proved that the constant factor is best possible. Recently, Yang  gave a half-discrete Hilbert inequality with multi-parameters, and  gave the following half-discrete reverse Hilbert-type inequality with the best constant factor 4: For , , we have , and
In this paper, by using the way of weight functions and the Hermite-Hadamard inequality, a half-discrete reverse Mulholland-type inequality similar to (6) is given as follows:
Moreover, a best extension of (7) with multi-parameters, the equivalent forms and the relating homogeneous inequalities are considered.
2 Some lemmas
Lemma 1 If , , setting weight functions and as follows:
Proof Substituting of in (8), by calculation, we have
Since by the conditions and for fixed
is strictly decreasing and strictly convex in , then by the Hermite-Hadamard inequality (cf. ), we find
that is, (10) is valid. □
Lemma 2 Let the assumptions of Lemma 1 be fulfilled and, additionally, , , , , is a non-negative measurable function in . Then we have the following inequalities:
Proof By the reverse Hölder inequality (cf. ) and (10), it follows that
Then, by the Lebesgue term-by-term integration theorem (cf. ), we have
and (11) follows. Still, by the reverse Hölder inequality, we have
Then, by the Lebesgue term-by-term integration theorem, we have
and then in view of (10), inequality (12) follows. □
3 Main results
In this paper, for (), we still use the normal expressions and . We also introduce two functions
wherefrom and .
Theorem 1 If , , , , , and , then we have the following equivalent inequalities:
where the constant is best possible.
Proof By the Lebesgue term-by-term integration theorem, there are two expressions for I in (13). In view of (11), for , we have (14). By the reverse Hölder inequality, we have
Then by (14) we have (13). On the other hand, assuming that (13) is valid, setting
we obtain that . By (11), we find . If , then (14) is trivially valid; if , then by (13) we have
that is, (14) is equivalent to (13). In view of (12), for
we have (15). By the reverse Hölder inequality, we find
Then by (15) we have (13). On the other hand, assuming that (13) is valid, setting
we obtain that . By (12), we find . If , then (15) is trivially valid; if , then by (13) we have
that is, (15) is equivalent to (13). Hence, inequalities (13), (14) and (15) are equivalent.
For , set , ; , , and
If there exists a positive number k () such that (13) is valid when replacing with k, then, in particular, it follows that
Hence by (18) and (19) it follows that
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. □
Remark 1 (i) For , , in (13), (14) and (15), we have (7) and the following equivalent inequalities:
(ii) Setting , and
in (13), by simplifications, we find the following inequality with the homogeneous kernel:
It is evident that (22) is equivalent to (13), and then the constant factor in (22) is still best possible. In the same way as in (14) and (15), we have the following inequalities equivalent to (13) with the best constant factor :
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This work is supported by the National Natural Science Foundation of China (No. 61370186).
The authors declare that they have no competing interests.
TL participated in the design of the study and performed the numerical analysis. BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. LH conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Liu, T., Yang, B. & He, L. On a half-discrete reverse Mulholland-type inequality and an extension. J Inequal Appl 2014, 103 (2014). https://doi.org/10.1186/1029-242X-2014-103
- Mulholland-type inequality
- weight function
- equivalent form