Abstract
In the present paper, we prove some equivalent statements of a Hilbert-type integral inequality in the whole plane with intermediate variables. In our theorems, the constant factor is associated to the Hurwitz zeta function and we prove that it is the best possible. We also derive various special cases and applications.
On the occasion of the 100th birthday of Tosio Kato
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References
Schur, I.: Bernerkungen sur Theorie der beschrankten Billnearformen mit unendlich vielen Veranderlichen. J. Math. 140, 1–28 (1911)
Hardy, G.H.: Note on a theorem of Hilbert concerning series of positive terms. Proc. Lond. Math. Soc. 23(2) (1925). Records of Proc. xlv-xlvi
Hardy, G.H., Littlewood, J.E., P\(\acute{o}\)lya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Yang, B.C.: On Hilbert’s integral inequality. J. Math. Anal. Appl. 220, 778–785 (1998)
Yang, B.C.: A note on Hilbert’s integral inequality. Chin. Q. J. Math. 13(4), 83–86 (1998)
Yang, B.C.: On an extension of Hilbert’s integral inequality with some parameters. Aust. J. Math. Anal. Appl. 1(1), 1–8 (2004). Art. 11
Yang, B.C., Brneti\(\acute{c}\), I., Krni\(\acute{c}\), M., Pe\( \breve{c}\)ari\(\acute{c}\), J.E.: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal. Appl. 8(2), 259–272 (2005)
Krni\(\acute{c}\), M., Pe\(\breve{c}\)ari\(\acute{c}\), J.E.: Hilbert’s inequalities and their reverses. Publ. Math. Debrecen 67(3–4), 315–331 (2005)
Hong, Y.: On Hardy-Hilbert integral inequalities with some parameters. J. Inequal. Pure Appl. Math. 6(4), 1–10 (2005). Art. 92
Arpad, B., Choonghong, O.: Best constant for certain multi linear integral operator. J. Inequal. Appl. 28582 (2006)
Li, Y.J., He, B.: On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 76(1), 1–13 (2007)
Zhong, W.Y., Yang, B.C.: On multiple Hardy-Hilbert’s integral inequality with kernel. J. Inequal. Appl. 2007, 17 pp. (2007). Art. ID 27962
Xu, J.S.: Hardy-Hilbert’s Inequalities with two parameters. Adv. Math. 36(2), 63–76 (2007)
Rassias, M.Th., Yang, B.C.: On a multidimensional half - discrete Hilbert - type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 242, 800–813 (2014)
Rassias, M.Th., Yang, B.C.: A Hilbert - type integral inequality in the whole plane related to the hyper geometric function and the beta function. J. Math. Anal. Appl. 428(2), 1286–1308 (2015)
Rassias, M.Th., Yang, B.C.: On a Hardy-Hilbert-type inequality with a general homogeneous kernel. Int. J. Nonlinear Anal. Appl. 7(1), 249–269 (2016)
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)
Kato, T.: Variation of discrete spectra. Commun. Math. Phys. 111(3), 501–504 (1987)
Kato, T., Satake, I.: An algebraic theory of Landau-Kolmogorov inequalities. Tohoku Math. J. 33(3), 421–428 (1981)
Kato, T.: On an inequality of Hardy, Littlewood, and P\(\acute{o}\)lya. Adv. Math. 7, 217–218 (1971)
Kato, T.: Demicontinuity, hemicontinuity and monotonicity, II. Bull. Am. Math. Soc. 73, 886–889 (1967)
Yang, B.C.: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)
Yang, B.C.: Hilbert-Type Integral Inequalities. Bentham Science Publishers Ltd., The United Emirates (2009)
Yang, B.C.: On Hilbert-type integral inequalities and their operator expressions. J. Guangaong Univ. Edu. 33(5), 1–17 (2013)
Yang, B.C.: A new Hilbert-type integral inequality. Soochow J. Math. 33(4), 849–859 (2007)
He, B., Yang, B.C.: On a Hilbert-type integral inequality with the homogeneous kernel of 0-degree and the hypergeometric function. Math. Pract. Theory 40(18), 105–211 (2010)
Yang, B.C.: A new Hilbert-type integral inequality with some parameters. J. Jilin Univ. (Science Edition) 46(6), 1085–1090 (2008)
Zeng, Z., Xie, Z.T.: On a new Hilbert-type integral inequality with the homogeneous kernel of degree 0 and the integral in whole plane. J. Inequal. Appl. 2010, 9 pp. (2010). Article ID 256796
Wang, A.Z., Yang, B.C.: A new Hilbert-type integral inequality in whole plane with the non-homogeneous kernel. J. Inequal. Appl. 2011, 123 (2011)
Xin, D.M., Yang, B.C.: A Hilbert-type integral inequality in whole plane with the homogeneous kernel of degree -2. J. Inequal. Appl. 2011, 11 pp. (2011). Article ID 401428
He, B., Yang, B.C.: On an inequality concerning a non-homogeneous kernel and the hypergeometric function. Tamsul Oxf. J. Inf. Math. Sci. 27(1), 75–88 (2011)
Xie, Z.T., Zeng, Z., Sun, Y.F.: A new Hilbert-type inequality with the homogeneous kernel of degree -2. Adv. Appl. Math. Sci. 12(7), 391–401 (2013)
Huang, Q.L., Wu, S.H., Yang, B.C.: Parameterized Hilbert-type integral inequalities in the whole plane. Sci. World J. 2014, 8 pp. (2014). Article ID 169061
Zheng, 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. 3(1), 11–20 (2014)
Huang, X.Y., Cao, J.F., He, B., Yang, B.C.: Hilbert-type and Hardy-type integral inequalities with operator expressions and the best constants in the whole plane. J. Inequal. Appl. 2015, 129 (2015)
Gu, Z.H., Yang, B.C.: A Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a few parameters. J. Inequal. Appl. 2015, 314 (2015)
Hong, Y.: On the structure character of Hilbert’s type integral inequality with homogeneous kernal and applications. J. Jilin Univ. (Science Edition) 55(2), 189–194 (2017)
Rassias, M.Th., Yang, B.C.: Equivalent properties of a Hilbert-type integral inequality with the best constant factor related the Hurwitz zeta function. Ann. Funct. Anal. 9(2), 282–295 (2018)
Hong, Y., Huang, Q.L., Yang, B.C., Liao, J.Q.: 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, 316 (2017)
Yang, B.C., Chen, Q.: Equivalent conditions of existence of a class of reverse Hardy-type integral inequalities with nonhomogeneous kernel. J. Jilin Univ. (Science Edition) 55(4), 804–808 (2017)
Yang, B.C.: Equivalent conditions of the existence of Hardy-type and Yang-Hilbert-type integral inequalities with nonhomogeneous kernel. J. Guangdong Univ. Edu. 37(3), 5–10 (2017)
Yang, B.C.: On some equivalent conditions related to the bounded property of Yang-Hilbert-type operator. J. Guangdong Univ. Edu. 37(5), 5–11 (2017)
Yang, Z.M., Yang, B.C.: Equivalent conditions of the existence of the reverse Hardy-type integral inequalities with the nonhomogeneous kernel. J. Guangdong Univ. Edu. 37(5), 28–32 (2017)
Kuang, J.C.: Real and Functional Analysis (Continuation) (second volume). Higher Education Press, Beijing (2015)
Wang, J.Q., Guo, D.R.: Introduction to Special Functions. Science Press, Beijing (1979)
Kuang, J.C.: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004)
Acknowledgements
This work is supported by the National Natural Science Foundation (No. 617721-40), and Science and Technology Planning Project Item of Guangzhou City (No. 201707010229). We are grateful for their support.
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Rassias, T.M., Yang, B. (2019). On a Few Equivalent Statements of a Hilbert-Type Integral Inequality in the Whole Plane with the Hurwitz Zeta Function. In: Rassias, T.M., Zagrebnov, V.A. (eds) Analysis and Operator Theory . Springer Optimization and Its Applications, vol 146. Springer, Cham. https://doi.org/10.1007/978-3-030-12661-2_15
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DOI: https://doi.org/10.1007/978-3-030-12661-2_15
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