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On a half-discrete Hilbert-type inequality similar to Mulholland’s inequality
Journal of Inequalities and Applications volume 2013, Article number: 290 (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.
Assuming that , , , we have the following Hilbert integral inequality (cf. ):
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  gave an extension of (1). By 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 with , , , , , , then
where the constant factor is the best possible. Moreover, if is finite and () is decreasing for (), then for , , , , we have
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 . But they did not prove that the constant factors in the inequalities are the best possible. Moreover, Yang  gave an inequality with the particular kernel and an interval variable, and proved that the constant factor is the best possible. Recently,  and  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:
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. ), 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. ) and (10), it follows
Then by the Lebesgue term-by-term integration theorem (cf. ), 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
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 .
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:
Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.
Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Acaremic, Boston; 1991.
Yang B: Hilbert-Type Integral Inequalities. Bentham Science Publishers, Sharjah; 2009.
Yang B: Discrete Hilbert-Type Inequalities. Bentham Science Publishers, Sharjah; 2011.
Yang B: On a new extension of Hilbert’s inequality with some parameters. Acta Math. Hung. 2005, 108(4):337–350. 10.1007/s10474-005-0229-4
Yang B: On Hilbert’s integral inequality. J. Math. Anal. Appl. 1998, 220: 778–785. 10.1006/jmaa.1997.5877
Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijin; 2009.
Yang B, Brnetić I, Krnić M, Pečarić J: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal. Appl. 2005, 8(2):259–272.
Krnić M, Pečarić J: Hilbert’s inequalities and their reverses. Publ. Math. (Debr.) 2005, 67(3–4):315–331.
Jin J, Debnath L: On a Hilbert-type linear series operator and its applications. J. Math. Anal. Appl. 2010, 371: 691–704. 10.1016/j.jmaa.2010.06.002
Azar L: On some extensions of Hardy-Hilbert’s inequality and applications. J. Inequal. Appl. 2009., 2009: Article ID 546829
Yang B, Rassias TM: On the way of weight coefficient and research for Hilbert-type inequalities. Math. Inequal. Appl. 2003, 6(4):625–658.
Arpad B, Choonghong O: Best constant for certain multilinear integral operator. J. Inequal. Appl. 2006., 2006: Article ID 28582
Kuang J, Debnath L: On Hilbert’s type inequalities on the weighted Orlicz spaces. Pac. J. Appl. Math. 2007, 1(1):95–103.
Zhong W: The Hilbert-type integral inequality with a homogeneous kernel of − λ -degree. J. Inequal. Appl. 2008., 2008: Article ID 917392
Li Y, He B: On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 2007, 76(1):1–13. 10.1017/S0004972700039423
Yang B: A mixed Hilbert-type inequality with a best constant factor. Int. J. Pure Appl. Math. 2005, 20(3):319–328.
Yang B: A half-discrete Hilbert’s inequality. J. Guangdong Univ. Educ. 2011, 31(3):1–7.
Yang B, Chen Q: A half-discrete Hilbert-type inequality with a homogeneous kernel and an extension. J. Inequal. Appl. 2011., 2011: Article ID 124. doi:10.1186/1029–242X-2011–124
Kuang J: Applied Inequalities. Shangdong Science Technic Press, Jinan; 2004.
Kuang J: Introduction to Real Analysis. Hunan Education Press, Chansha; 1996.
This work is supported by 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).
The authors declare that they have no competing interests.
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
- Hilbert-type inequality
- weight function
- equivalent form