Abstract
In this chapter, we establish some new generalizations of Azar’s results, which are relations between the Hardy–Hilbert inequality and the Gabriel inequality. As an application, we obtain a sharper form of the general Hardy-Hilbert inequality. The integral analogues of our main results are also given. Some Gabriel-type inequalities are also considered.
Mathematics Subject Classification (2000): Primary 26D15, Secondary 05E05
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References
Azar, L.E.: A relation between Hilbert and Carlson inequalities. J. Inequal. Appl. 2012, Art. 277 (2012)
Carlson, F.: Une inégalité. Ark. Mat. Astr. Fys. 25B(1), 1–5 (1935)
Das, N., Sahoo, S.: On a generalization of Hardy-Hilbert’s integral inequality. Bull. Acad. Ştiinţ. Repub. Mold. Mat. 60(2), 90–110 (2009)
Das, N., Sahoo, S.: A generalization of Hardy-Hilbert’s inequality for non-homogeneous kernel. Bull. Acad. Ştiinţ. Repub. Mold. Mat. 67(3), 29–44 (2011)
Gabriel, R.M.: An extension of an inequality due to Carlson. J. Lond. Math. Soc. 12, 130–132 (1937)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1952)
Krnić, M., Pečarić, J.: General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 8(1), 29–51 (2005)
Krnić, M., Pečarić, J., Vuković, J.: Discrete Hilbert-type inequalities with general homogeneous kernels. Rend. Circ. Mat. Palermo 60(1–2), 161–171 (2011)
Larson, L., Maligranda, L., Pečarić, J., Persson, L.E.: Multiplicative Inequalities of Carlson Type and Interpolation. World Scientific, Hackensack (2006)
Mitrinovic, D.S., Pecaric, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)
Acknowledgements
We would like to express our gratitude to Prof. Mario Krnić for reading the manuscript and for his very helpful remarks.
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Adiyasuren, V., Batbold, T. (2014). On a Relation Between the Hardy–Hilbert and Gabriel Inequalities. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 95. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1246-9_1
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DOI: https://doi.org/10.1007/978-1-4939-1246-9_1
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