On a Reverse Half-Discrete Hardy-Hilbert’s Inequality with Parameters
Abstract
:1. Introduction
2. Some Lemmas
3. Main Results
- (i)
- is independent of;
- (ii)
- is expressible as a single integral;
- (iii)
- is the best possible of (6);
- (iv)
- Ifthen.
4. Two Corollaries and Some Particular Inequalities
- (i)
- is independent of;
- (ii)
- is expressible as a single integral;
- (iii)
- is the best possible of (17);
- (iv)
- Ifthen we have.
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Hardy, G.H.; Littlewood, J.E.; Polya, G. Inequalities; Cambridge University Press: Cambridge, UK, 1934. [Google Scholar]
- Yang, B.C. The Norm of Operator and Hilbert-Type Inequalities; Science Press: Beijing, China, 2009. [Google Scholar]
- Yang, B.C. Hilbert-Type Integral Inequalities; Bentham Science Publishers Ltd.: Sharjah, UAE, 2009. [Google Scholar]
- Yang, B.C. On the norm of an integral operator and applications. J. Math. Anal. Appl. 2006, 321, 182–192. [Google Scholar] [CrossRef] [Green Version]
- Xu, J.S. Hardy-Hilbert’s inequalities with two parameters. Adv. Math. 2007, 36, 63–76. [Google Scholar]
- Yang, B.C. On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 2007, 325, 529–541. [Google Scholar] [CrossRef]
- Xie, Z.T.; Zeng, Z.; Sun, Y.F. A new Hilbert-type inequality with the homogeneous kernel of degree −2. Adv. Appl. Math. Sci. 2013, 12, 391–401. [Google Scholar]
- Zhen, Z.; Raja Rama Gandhi, K.; Xie, Z.T. A new Hilbert-type inequality with the homogeneous kernel of degree −2 and with the integral. Bull. Math. Sci. Appl. 2014, 3, 11–20. [Google Scholar]
- Xin, D.M. A Hilbert-type integral inequality with the homogeneous kernel of zero degree. Math. Theory Appl. 2010, 30, 70–74. [Google Scholar]
- Azar, L.E. The connection between Hilbert and Hardy inequalities. J. Inequal. Appl. 2013, 452, 2013. [Google Scholar] [CrossRef]
- Batbold, T.; Sawano, Y. Sharp bounds for m-linear Hilbert-type operators on the weighted Morrey spaces. Math. Inequal. Appl. 2017, 20, 263–283. [Google Scholar] [CrossRef]
- Adiyasuren, V.; Batbold, T.; Krnic, M. Multiple Hilbert-type inequalities involving some differential operators. Banach. J. Math. Anal. 2016, 10, 320–337. [Google Scholar] [CrossRef]
- Adiyasuren, V.; Batbold, T.; Krnic, M. Hilbert–type inequalities involving differential operators, the best constants and applications. Math. Inequal. Appl. 2015, 18, 111–124. [Google Scholar] [CrossRef]
- Rassias, M.T.H.; Yang, B.C. On half-discrete Hilbert’s inequality. Appl. Math. Comput. 2013, 220, 75–93. [Google Scholar] [CrossRef]
- Yang, B.C.; Krnic, M. A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0. J. Math. Inequal. 2012, 6, 401–417. [Google Scholar]
- Rassias, M.T.H.; Yang, B.C. A multidimensional half – discrete Hilbert–type inequality and the Riemann zeta function. Appl. Math. Comput. 2013, 225, 263–277. [Google Scholar] [CrossRef]
- Rassias, M.T.H.; Yang, B.C. On a multidimensional half–discrete Hilbert–type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 2013, 242, 800–813. [Google Scholar] [CrossRef]
- Huang, Z.X.; Yang, B.C. On a half-discrete Hilbert-Type inequality similar to Mulholland’s inequality. J. Inequal. Appl. 2013, 290, 2013. [Google Scholar] [CrossRef]
- Yang, B.C.; Lebnath, L. Half-Discrete Hilbert-Type Inequalities; World Scientific Publishing: Singapore, 2014. [Google Scholar]
- Krnic, M.; Pecaric, J. Extension of Hilbert’s inequality. J. Math. Anal. Appl. 2006, 324, 150–160. [Google Scholar] [CrossRef]
- Adiyasuren, V.; Batbold, T.; Azar, L.E. A new discrete Hilbert-type inequality involving partial sums. J. Inequal. Appl. 2019, 127, 2019. [Google Scholar] [CrossRef]
- Hong, Y.; Wen, Y. A necessary and Sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor. Ann. Math. 2016, 37, 329–336. [Google Scholar]
- Hong, Y. On the structure character of Hilbert’s type integral inequality with homogeneous kernel and applications. J. Jilin Univ. Sci. Ed. 2017, 55, 189–194. [Google Scholar]
- Hong, Y.; Huang, Q.L.; Yang, B.C.; Liao, J.L. The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications. J. Inequal. Appl. 2017, 2017, 316. [Google Scholar] [CrossRef]
- Xin, D.M.; Yang, B.C.; Wang, A.Z. Equivalent property of a Hilbert-type integral inequality related to the beta function in the whole plane. J. Funct. Spaces 2018. [Google Scholar] [CrossRef]
- Hong, Y.; He, B.; Yang, B.C. Necessary and Sufficient Conditions for the Validity of Hilbert Type Integral Inequalities with a Class of Quasi-Homogeneous Kernels and Its Application in Operator Theory. J. Math. Inequal. 2018, 12, 777–788. [Google Scholar] [CrossRef]
- Kuang, J.C. Applied Inequalities; Shangdong Science and Technology Press: Jinan, China, 2004. [Google Scholar]
- Kuang, J.C. Real and Functional Analysis (Continuation); Higher Education Press: Beijing, China, 2015. [Google Scholar]
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Yang, B.; Wu, S.; Wang, A. On a Reverse Half-Discrete Hardy-Hilbert’s Inequality with Parameters. Mathematics 2019, 7, 1054. https://doi.org/10.3390/math7111054
Yang B, Wu S, Wang A. On a Reverse Half-Discrete Hardy-Hilbert’s Inequality with Parameters. Mathematics. 2019; 7(11):1054. https://doi.org/10.3390/math7111054
Chicago/Turabian StyleYang, Bicheng, Shanhe Wu, and Aizhen Wang. 2019. "On a Reverse Half-Discrete Hardy-Hilbert’s Inequality with Parameters" Mathematics 7, no. 11: 1054. https://doi.org/10.3390/math7111054
APA StyleYang, B., Wu, S., & Wang, A. (2019). On a Reverse Half-Discrete Hardy-Hilbert’s Inequality with Parameters. Mathematics, 7(11), 1054. https://doi.org/10.3390/math7111054