Abstract
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.
MSC:26D15.
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1 Introduction
Assuming that , , , we have the following Hilbert integral inequality (cf. [1]):
where the constant factor π is the 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 (cf. [1, 5]):
In 1998, by introducing an independent parameter , Yang [6] gave an extension of (1). By generalizing the results from [6], Yang [7] gave some best extensions of (1) and (2) as follows: If , , , is a non-negative homogeneous function of degree −λ with , , , , , , then
where the constant factor is the best possible. Moreover, if is finite and () is 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 about Hilbert-type inequalities are provided by [5, 8–16].
On the topic of half-discrete Hilbert-type inequalities with the general non-homogeneous kernels, Hardy et al. provided a few results in Theorem 351 of [1]. But they did not prove that the constant factors in the inequalities are the best possible. Moreover, Yang [17] gave an inequality with the particular kernel and an interval variable, and proved that the constant factor is the best possible. Recently, [18] and [19] gave the following half-discrete Hilbert inequality with the best constant factor π:
In this paper, by using the way of weight functions and Hadamard’s inequality, a half-discrete Hilbert-type inequality similar to (3) and (6) with the best constant factor is given as follows:
Moreover, the best extension of (7) with multi-parameters, some equivalent forms as well as the operator expressions are considered.
2 Some lemmas
Lemma 1 If , , setting weight functions and as follows:
we have
Proof Substitution of in (8), by calculation, yields
Since, for fixed and in view of the conditions,
is decreasing and strictly convex for , then by Hadamard’s inequality (cf. [20]), we find
namely, (10) follows. □
Lemma 2 Let the assumptions of Lemma 1 be fulfilled and, additionally, let , , , , be a non-negative measurable function in . Then we have the following inequalities:
Proof By Hölder’s inequality (cf. [20]) and (10), it follows
Then by the Lebesgue term-by-term integration theorem (cf. [21]), we have
and (11) follows. Still by Hölder’s 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
We introduce two functions
wherefrom, , and .
Theorem 1 If , , , , , , , and , then we have the following equivalent inequalities:
where the constant is the best possible in the above inequalities.
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 Hölder’s inequality, we have
Then by (14), we have (13). On the other hand, assuming that (13) is valid, setting
then . 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 Hölder’s inequality, we find
Then by (15), we have (13). On the other hand, assuming that (13) is valid, setting
then . 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 , setting , ; , , and , , if there exists a positive number k () such that (13) is valid as we replace with k, then, in particular, it follows
We find
and then (). Hence by (18) and (19), it follows
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. □
Remark 1 (i) Define the first type half-discrete Hilbert-type operator as follows: For , we define , satisfying
Then by (14) it follows , and then is a bounded operator with . Since by Theorem 1 the constant factor in (14) is the best possible, we have .
-
(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 .
Remark 2 For , , , in (13), (14) and (15), we have (7) and the following equivalent inequalities:
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Acknowledgements
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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ZH wrote and reformed the article. BY conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Huang, Z., Yang, B. On a half-discrete Hilbert-type inequality similar to Mulholland’s inequality. J Inequal Appl 2013, 290 (2013). https://doi.org/10.1186/1029-242X-2013-290
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DOI: https://doi.org/10.1186/1029-242X-2013-290