- Open Access
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.
- Mulholland-type inequality
- weight function
- equivalent form
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.
that is, (10) is valid. □
and then in view of (10), inequality (12) follows. □
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).
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.Google Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston; 1991.View ArticleGoogle 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: An extension of Mulholland’s inequality. Jordan J. Math. Stat. 2010,3(3):151-157.Google Scholar
- Yang B: On Hilbert’s integral inequality. J. Math. Anal. Appl. 1998, 220: 778-785. 10.1006/jmaa.1997.5877MathSciNetView ArticleGoogle Scholar
- Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing; 2009. (in Chinese)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.MathSciNetGoogle Scholar
- Krnić M, Pečarić J: Hilbert’s inequalities and their reverses. Publ. Math. (Debr.) 2005,67(3-4):315-331.Google 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 ArticleGoogle 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 T: On the way of weight coefficient and research for Hilbert-type inequalities. Math. Inequal. Appl. 2003,6(4):625-658.MathSciNetGoogle 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.MathSciNetGoogle 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 ArticleGoogle Scholar
- Yang B: A mixed Hilbert-type inequality with a best constant factor. Int. J. Pure Appl. Math. 2005,20(3):319-328.MathSciNetGoogle Scholar
- Yang B: A half-discrete Hilbert’s inequality. J. Guangdong Univ. Educ. 2011,31(3):1-7.Google Scholar
- Yang B: A half-discrete reverse Hilbert-type inequality with a homogeneous kernel of positive degree. J. Zhanjiang Norm. Coll. 2011,32(3):5-9.Google Scholar
- Kuang J: Applied Inequalities. Shangdong Science Technic Press, Jinan; 2004. (in Chinese)Google Scholar
- Kuang J: Introduction to Real Analysis. Hunan Education Press, Chansha; 1996. (in Chinese)Google Scholar
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