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2014 | OriginalPaper | Buchkapitel

Hilbert-Type Inequalities Including Some Operators, the Best Possible Constants and Applications: A Survey

verfasst von : Vandanjav Adiyasuren, Tserendorj Batbold, Mario Krnić

Erschienen in: Topics in Mathematical Analysis and Applications

Verlag: Springer International Publishing

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Abstract

The present work is a review article about some recent results dealing with Hilbert-type inequalities including certain operators in both integral and discrete case. A particular emphasis is given to inequalities including classical means operators. The constants appearing in all discussed inequalities are the best possible. For an illustration, some proofs are given, as well as some applications.

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Literatur
1.
Zurück zum Zitat Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York (1972) Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York (1972)
2.
Zurück zum Zitat Adiyasuren, V., Batbold, Ts.: Some new inequalities of Hardy-Hilbert type. Mong. Math. J. 15, 12–19 (2011) Adiyasuren, V., Batbold, Ts.: Some new inequalities of Hardy-Hilbert type. Mong. Math. J. 15, 12–19 (2011)
3.
Zurück zum Zitat Adiyasuren, V., Batbold, Ts.: Some new inequalities similar to Hilbert-type integral inequality with a homogeneous kernel. J. Math. Inequal. 6, 183–193 (2012) Adiyasuren, V., Batbold, Ts.: Some new inequalities similar to Hilbert-type integral inequality with a homogeneous kernel. J. Math. Inequal. 6, 183–193 (2012)
4.
Zurück zum Zitat Adiyasuren, V., Batbold, Ts.: New multiple inequality similar to Hardy-Hilbert’s integral inequality. Tamsui Oxf. J. Inf. Math. Sci. 28, 281–292 (2012) Adiyasuren, V., Batbold, Ts.: New multiple inequality similar to Hardy-Hilbert’s integral inequality. Tamsui Oxf. J. Inf. Math. Sci. 28, 281–292 (2012)
5.
Zurück zum Zitat Adiyasuren, V., Batbold, Ts., Krnić, M.: On several new Hilbert-type inequalities involving means operators. Acta Math. Sin. Engl. Ser. 29, 1493–1514 (2013) Adiyasuren, V., Batbold, Ts., Krnić, M.: On several new Hilbert-type inequalities involving means operators. Acta Math. Sin. Engl. Ser. 29, 1493–1514 (2013)
6.
Zurück zum Zitat Adiyasuren, V., Batbold, Ts., Krnić, M.: The best constants in multidimensional Hilbert-type inequalities involving some weighted means operators. Bull. Malays. Math. Sci. Soc. (in press) Adiyasuren, V., Batbold, Ts., Krnić, M.: The best constants in multidimensional Hilbert-type inequalities involving some weighted means operators. Bull. Malays. Math. Sci. Soc. (in press)
7.
Zurück zum Zitat Adiyasuren, V., Batbold, Ts., Krnić, M.: Hilbert-type inequalities involving differential operators, the best constants, and applications. Math. Inequal. Appl. (in press) Adiyasuren, V., Batbold, Ts., Krnić, M.: Hilbert-type inequalities involving differential operators, the best constants, and applications. Math. Inequal. Appl. (in press)
8.
Zurück zum Zitat Adiyasuren, V., Batbold, Ts., Krnić, M.: Half-discrete Hilbert-type inequalities with means operators, the best constants, and applications. Appl. Math. Comput. 231, 148–159 (2014) Adiyasuren, V., Batbold, Ts., Krnić, M.: Half-discrete Hilbert-type inequalities with means operators, the best constants, and applications. Appl. Math. Comput. 231, 148–159 (2014)
9.
Zurück zum Zitat Azar, L.E.: Two new forms of Hilbert-type integral inequality. Math. Inequal. Appl. 17, 937–946 (2014)MathSciNetMATH Azar, L.E.: Two new forms of Hilbert-type integral inequality. Math. Inequal. Appl. 17, 937–946 (2014)MathSciNetMATH
10.
Zurück zum Zitat Brnetić, I., Krnić, M., Pečarić, J.: Multiple Hilbert and Hardy-Hilbert inequalities with non-conjugate parameters. Bull. Aust. Math. Soc. 71, 447–457 (2005)CrossRefMATH Brnetić, I., Krnić, M., Pečarić, J.: Multiple Hilbert and Hardy-Hilbert inequalities with non-conjugate parameters. Bull. Aust. Math. Soc. 71, 447–457 (2005)CrossRefMATH
11.
Zurück zum Zitat Čižmešija, A., Krnić, M., Pečarić, J.: General Hilbert’s inequality with non-conjugate parameters. Math. Inequal. Appl. 11, 237–269 (2008)MathSciNetMATH Čižmešija, A., Krnić, M., Pečarić, J.: General Hilbert’s inequality with non-conjugate parameters. Math. Inequal. Appl. 11, 237–269 (2008)MathSciNetMATH
12.
Zurück zum Zitat Das, N., Sahoo, S.: New inequalities similar to Hardy-Hilbert’s inequality. Turk. J. Math. 34, 153–165 (2010)MathSciNetMATH Das, N., Sahoo, S.: New inequalities similar to Hardy-Hilbert’s inequality. Turk. J. Math. 34, 153–165 (2010)MathSciNetMATH
13.
Zurück zum Zitat Das, N., Sahoo, S.: On a generalization of Hardy-Hilbert’s integral inequality. Bull. Acad. Sţiinţe Repub. Mold. Mat. 2, 91–110 (2010) Das, N., Sahoo, S.: On a generalization of Hardy-Hilbert’s integral inequality. Bull. Acad. Sţiinţe Repub. Mold. Mat. 2, 91–110 (2010)
14.
Zurück zum Zitat Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1967) Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1967)
15.
Zurück zum Zitat He, B., Yang, B.: On a half-discrete inequality with a generalized homogeneous kernel. J. Inequal. Appl. 2012:30, 15pp (2012) He, B., Yang, B.: On a half-discrete inequality with a generalized homogeneous kernel. J. Inequal. Appl. 2012:30, 15pp (2012)
16.
17.
Zurück zum Zitat Krnić, M., Pečarić, J.: General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 8, 29–52 (2005)MathSciNet Krnić, M., Pečarić, J.: General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 8, 29–52 (2005)MathSciNet
18.
Zurück zum Zitat Krnić, M., Mingzhe, G., Pečarić, J., Xuemei, G.: On the best constant in Hilbert’s inequality. Math. Inequal. Appl. 8, 317–329 (2005)MathSciNetMATH Krnić, M., Mingzhe, G., Pečarić, J., Xuemei, G.: On the best constant in Hilbert’s inequality. Math. Inequal. Appl. 8, 317–329 (2005)MathSciNetMATH
19.
Zurück zum Zitat Krnić, M., Pečarić, J., Perić, I., Vuković, P.: Recent Advances in Hilbert-Type Inequalities. Element, Zagreb (2012) Krnić, M., Pečarić, J., Perić, I., Vuković, P.: Recent Advances in Hilbert-Type Inequalities. Element, Zagreb (2012)
20.
Zurück zum Zitat Krnić, M., Pečarić, J., Vuković, P.: A unified treatment of half-discrete Hilbert-type inequalities with a homogeneous kernel. Mediterr. J. Math. 10, 1697–1716 (2013)MathSciNetCrossRefMATH Krnić, M., Pečarić, J., Vuković, P.: A unified treatment of half-discrete Hilbert-type inequalities with a homogeneous kernel. Mediterr. J. Math. 10, 1697–1716 (2013)MathSciNetCrossRefMATH
21.
Zurück zum Zitat Kufner, A., Maligranda, L., Persson, L.E.: The Hardy Inequality – About its History and Some Related Results. Vydavatelský servis, Pilsen (2007)MATH Kufner, A., Maligranda, L., Persson, L.E.: The Hardy Inequality – About its History and Some Related Results. Vydavatelský servis, Pilsen (2007)MATH
22.
Zurück zum Zitat Liu, X., Yang, B.: On a new Hilbert-Hardy-type integral operator and applications. J. Inequal. Appl. 2010:812636, 10pp (2010) Liu, X., Yang, B.: On a new Hilbert-Hardy-type integral operator and applications. J. Inequal. Appl. 2010:812636, 10pp (2010)
23.
Zurück zum Zitat Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht/Boston/London (1993)CrossRefMATH Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht/Boston/London (1993)CrossRefMATH
24.
Zurück zum Zitat Perić, I., Vuković, P.: Hardy-Hilbert’s inequalities with a general homogeneous kernel. Math. Inequal. Appl. 12, 525–536 (2009)MathSciNet Perić, I., Vuković, P.: Hardy-Hilbert’s inequalities with a general homogeneous kernel. Math. Inequal. Appl. 12, 525–536 (2009)MathSciNet
25.
Zurück zum Zitat Perić, I., Vuković, P.: Multiple Hilbert’s type inequalities with a homogeneous kernel. Banach J. Math. Anal. 5, 33–43 (2011)MathSciNetCrossRefMATH Perić, I., Vuković, P.: Multiple Hilbert’s type inequalities with a homogeneous kernel. Banach J. Math. Anal. 5, 33–43 (2011)MathSciNetCrossRefMATH
26.
27.
Zurück zum Zitat Rassias, M.Th., Yang, B.: A multidimensional Hilbert–type integral inequality related to the Riemann zeta function. In: Daras, N. (ed.) Applications of Mathematics and Informatics to Science and Engineering. Springer, New York 417–433 (2014) Rassias, M.Th., Yang, B.: A multidimensional Hilbert–type integral inequality related to the Riemann zeta function. In: Daras, N. (ed.) Applications of Mathematics and Informatics to Science and Engineering. Springer, New York 417–433 (2014)
28.
Zurück zum Zitat Sulaiman, W.T.: On three inequalities similar to Hardy-Hilbert’s integral inequality. Acta Math. Univ. Comenianae LXXVI, 273–278 (2007) Sulaiman, W.T.: On three inequalities similar to Hardy-Hilbert’s integral inequality. Acta Math. Univ. Comenianae LXXVI, 273–278 (2007)
29.
Zurück zum Zitat Sulaiman, W.T.: On two new inequalities similar to Hardy-Hilbert’s integral inequality. Soochow J. Math. 33, 497–501 (2007)MathSciNetMATH Sulaiman, W.T.: On two new inequalities similar to Hardy-Hilbert’s integral inequality. Soochow J. Math. 33, 497–501 (2007)MathSciNetMATH
30.
Zurück zum Zitat Sulaiman, W.T.: On two new inequalities similar to Hardy-Hilbert’s integral inequality. Int. J. Math. Anal. 4(37–40), 1823–1828 (2010)MathSciNetMATH Sulaiman, W.T.: On two new inequalities similar to Hardy-Hilbert’s integral inequality. Int. J. Math. Anal. 4(37–40), 1823–1828 (2010)MathSciNetMATH
31.
Zurück zum Zitat Yang, B.: On a Hardy-Carleman’s type inequality. Taiwan. J. Math. 9, 469–475 (2005)MATH Yang, B.: On a Hardy-Carleman’s type inequality. Taiwan. J. Math. 9, 469–475 (2005)MATH
32.
Zurück zum Zitat Yang, B.: A half-discrete Hilbert-type inequality with a non-homogeneous kernel and two variables. Mediterr. J. Math. 10, 677–692 (2013)MathSciNetCrossRefMATH Yang, B.: A half-discrete Hilbert-type inequality with a non-homogeneous kernel and two variables. Mediterr. J. Math. 10, 677–692 (2013)MathSciNetCrossRefMATH
33.
Zurück zum Zitat Yang, B., Krnić, M.: A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0. J. Math. Inequal. 6, 401–417 (2012)MathSciNetMATH Yang, B., Krnić, M.: A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0. J. Math. Inequal. 6, 401–417 (2012)MathSciNetMATH
34.
Metadaten
Titel
Hilbert-Type Inequalities Including Some Operators, the Best Possible Constants and Applications: A Survey
verfasst von
Vandanjav Adiyasuren
Tserendorj Batbold
Mario Krnić
Copyright-Jahr
2014
DOI
https://doi.org/10.1007/978-3-319-06554-0_2

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