- Open Access
On a half-discrete Hilbert-type inequality similar to Mulholland’s inequality
© Huang and Yang; licensee Springer 2013
- Received: 23 January 2013
- Accepted: 30 April 2013
- Published: 7 June 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.
- Hilbert-type inequality
- weight function
- equivalent form
Moreover, the best extension of (7) with multi-parameters, some equivalent forms as well as the operator expressions are considered.
namely, (10) follows. □
and then in view of (10), inequality (12) follows. □
wherefrom, , and .
where the constant is the best possible in the above inequalities.
that is, (15) is equivalent to (13). Hence, inequalities (13), (14) and (15) are equivalent.
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. □
- (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 .
This work is supported by 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.MATHGoogle Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Acaremic, Boston; 1991.View ArticleMATHGoogle Scholar
- Yang B: Hilbert-Type Integral Inequalities. Bentham Science Publishers, Sharjah; 2009.Google Scholar
- Yang B: Discrete Hilbert-Type Inequalities. Bentham Science Publishers, Sharjah; 2011.Google Scholar
- Yang B: On a new extension of Hilbert’s inequality with some parameters. Acta Math. Hung. 2005, 108(4):337–350. 10.1007/s10474-005-0229-4View ArticleMATHGoogle Scholar
- Yang B: On Hilbert’s integral inequality. J. Math. Anal. Appl. 1998, 220: 778–785. 10.1006/jmaa.1997.5877MathSciNetView ArticleMATHGoogle Scholar
- Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijin; 2009.Google Scholar
- Yang B, Brnetić I, Krnić M, Pečarić J: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal. Appl. 2005, 8(2):259–272.MathSciNetMATHGoogle Scholar
- Krnić M, Pečarić J: Hilbert’s inequalities and their reverses. Publ. Math. (Debr.) 2005, 67(3–4):315–331.MATHGoogle Scholar
- 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.2010.06.002MathSciNetView ArticleMATHGoogle Scholar
- Azar L: On some extensions of Hardy-Hilbert’s inequality and applications. J. Inequal. Appl. 2009., 2009: Article ID 546829Google Scholar
- Yang B, Rassias TM: On the way of weight coefficient and research for Hilbert-type inequalities. Math. Inequal. Appl. 2003, 6(4):625–658.MathSciNetMATHGoogle Scholar
- Arpad B, Choonghong O: Best constant for certain multilinear integral operator. J. Inequal. Appl. 2006., 2006: Article ID 28582Google Scholar
- Kuang J, Debnath L: On Hilbert’s type inequalities on the weighted Orlicz spaces. Pac. J. Appl. Math. 2007, 1(1):95–103.MathSciNetMATHGoogle Scholar
- Zhong W: The Hilbert-type integral inequality with a homogeneous kernel of − λ -degree. J. Inequal. Appl. 2008., 2008: Article ID 917392Google Scholar
- Li Y, He B: On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 2007, 76(1):1–13. 10.1017/S0004972700039423View ArticleMATHGoogle Scholar
- Yang B: A mixed Hilbert-type inequality with a best constant factor. Int. J. Pure Appl. Math. 2005, 20(3):319–328.MathSciNetMATHGoogle Scholar
- Yang B: A half-discrete Hilbert’s inequality. J. Guangdong Univ. Educ. 2011, 31(3):1–7.MATHGoogle Scholar
- 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-2011–124Google Scholar
- Kuang J: Applied Inequalities. Shangdong Science Technic Press, Jinan; 2004.Google Scholar
- Kuang J: Introduction to Real Analysis. Hunan Education Press, Chansha; 1996.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.