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On inequalities of Hilbert's type

  • Yongjin Li (a1) and Bing He (a2)

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By introducing the function 1/(min{x, y}), we establish several new inequalities similar to Hilbert's type inequality. Moreover, some further unification of Hardy-Hilbert's and Hardy-Hilbert's type integral inequality and its equivalent form with the best constant factor are proved, which contain the classic Hilbert's inequality as special case.

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References

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[1]Chow, Y.C., ‘On inequalities of Hilbert and Widder’, J. London Math. Soc. 14 (1939), 151154.
[2]Gao, M., ‘On Hilbert's inequality and its applications’, J. Math. Anal. Appl. 212 (1997), 316323.
[3]Hardy, G.H., ‘Note on a theorem of Hilbert’, Math. Z. 6 (1920), 314317.
[4]Hardy, G.H., Littlewood, J.E. and Polya, G., Inequalities (Cambridge University Press, Cambridge, 1934).
[5]Kuang, J., ‘Applied inequalities’, (in Chinese), Second edition (Hunan Education Press, Changsha).
[6]Kuang, J., ‘On new extensions of Hilbert's integral inequality’, Math. Anal. Appl. 235 (1999), 608614.
[7]Mitrinovic, D.S., Pecaric, J.E. and Fink, A.M., Inequalities involving functions and their integrals and derivatives (Kluwer Academic, Boston, 1991).
[8]Pachpatte, B.G., ‘Inequalities similar to the integral analogue of Hilbert's inequality’, Tamkang J. Math. 30 (1999), 139146.
[9]Sulaiman, W.T., ‘Four inequalities similar to Hardy-Hilbert's integral inequality’, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), Article 76, 8 pp.
[10]Yang, B., ‘An extension of Hardy-Hilbert's inequality’, (in Chinese), Chinese Ann. Math. Ser. A 23 (2002), 247254.
[11]Yang, B., ‘On the norm of an integral operator and applications’, J. Math. Anal. Appl. 321 (2006), 182192.
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On inequalities of Hilbert's type

  • Yongjin Li (a1) and Bing He (a2)

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