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A discrete Hilbert-type inequality in the whole plane. (English) Zbl 1335.26013

Summary: By the use of weight coefficients and technique of real analysis, a discrete Hilbert-type inequality in the whole plane with multi-parameters and a best possible constant factor is given. The equivalent forms, the operator expressions, and a few particular inequalities are considered.

MSC:

26D15 Inequalities for sums, series and integrals
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References:

[1] Hardy, GH: Note on a theorem of Hilbert concerning series of positive terms. Proc. Lond. Math. Soc. 23(2), 45-46 (1925)
[2] Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934) · Zbl 0010.10703
[3] Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht (1991) · Zbl 0744.26011
[4] Yang, BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)
[5] Yang, BC: Discrete Hilbert-Type Inequalities. Bentham Science Publishers, Sharjah (2011)
[6] Gao, MZ, Yang, BC: On extended Hilbert’s inequality. Proc. Am. Math. Soc. 126(3), 751-759 (1998) · Zbl 0935.26011
[7] Yang, BC, Debnath, L: On a new generalization of Hardy-Hilbert’s inequality and its applications. J. Math. Anal. Appl. 233, 484-497 (1999) · Zbl 0935.26009
[8] Yang, BC: On new generalizations of Hilbert’s inequality. J. Math. Anal. Appl. 248, 29-40 (2000) · Zbl 0970.26009
[9] Yang, BC, Debnath, L: On the extended Hardy-Hilbert’s inequality. J. Math. Anal. Appl. 272, 187-199 (2002) · Zbl 1009.26016
[10] Yang, BC: On a generalization of Hilbert’s double series theorem. Math. Inequal. Appl. 5(2), 197-204 (2002) · Zbl 1023.26012
[11] Yang, BC, Rasias, TM: On the way of weight coefficient and research for the Hilbert-type inequalities. Math. Inequal. Appl. 6(4), 625-658 (2003) · Zbl 1046.26012
[12] Yang, BC: On new extension of Hilbert’s inequality. Acta Math. Hung. 104(4), 291-299 (2004) · Zbl 1062.26023
[13] Yang, BC: On a new Hardy-Hilbert’s type inequality. Math. Inequal. Appl. 7(3), 355-363 (2004) · Zbl 1067.26024
[14] Krnić, M, Pečarić, JE: Hilbert’s inequalities and their reverses. Publ. Math. (Debr.) 67(3-4), 315-331 (2005) · Zbl 1092.26014
[15] Yang, BC: A new Hilbert-type inequality. Bull. Belg. Math. Soc. Simon Stevin 13(3), 479-487 (2006) · Zbl 1129.26024
[16] Yang, BC: On the norm of a self-adjoint operator and applications to Hilbert’s type inequalities. Bull. Belg. Math. Soc. Simon Stevin 13, 577-584 (2006) · Zbl 1128.47010
[17] Yang, BC: On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 325, 529-541 (2007) · Zbl 1114.47010
[18] Li, YJ, He, B: On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 76(1), 1-13 (2007) · Zbl 1135.26016
[19] Yang, BC, Krnić, M: On the norm of a multi-dimensional Hilbert-type operator. Sarajevo J. Math. 7(20), 223-243 (2011) · Zbl 1254.47011
[20] Krnić, M, Vuković, P: On a multidimensional version of the Hilbert-type inequality. Anal. Math. 38, 291-303 (2012) · Zbl 1274.26059
[21] Chen, Q, Yang, BC: A survey on the study of Hilbert-type inequalities. J. Inequal. Appl. 2015, 302 (2015) · Zbl 1336.26030
[22] Shi, YP, Yang, BC: On a multidimensional Hilbert-type inequality with parameters. J. Inequal. Appl. 2015, 371 (2015) · Zbl 1336.26044
[23] Shi, YP, Yang, BC: A new Hardy-Hilbert-type inequality with multiparameters and a best possible constant factor. J. Inequal. Appl. 2015, 380 (2015) · Zbl 1329.26042
[24] Yang, BC, Chen, Q: On a Hardy-Hilbert-type inequality with parameters. J. Inequal. Appl. 2015, 339 (2015) · Zbl 1336.26047
[25] Kuang, JC: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004)
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