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A Hilbert inequality and an Euler-Maclaurin summation formula

  • Mario Krnić and Josip Pečarić

Abstract

We obtain a generalized discrete Hilbert and Hardy-Hilbert inequality with non-conjugate parameters by means of an Euler-Maclaurin summation formula. We derive some general results for homogeneous functions and compare our findings with existing results. We improve some earlier results and apply the results to some special homogeneous functions.

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Copyright

References

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[1]Bicheng, Y., “On an extension of Hardy-Hilbert's inequality with some parameters”, Math. Inequal. Appl. (to appear).
[2]Bonsall, F. F., “Inequalities with non-conjugate parameters”, Quart. J. Math. Oxford (2) 2 (1951) 135150.
[3]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge Univ. Press, Cambridge, 1952).
[4]Jichang, K. and Debnath, L., “On new generalizations of Hilbert's inequality and their applications”, Math. Inequal. Appl. 245 (2000) 248265.
[5]Krnić, M. and Pečarić, J., “General Hilbert's and Hardy's inequalities”, Math. Inequal. Appl. 8 (2005) 2951.
[6]Krnić, M. and Pečarić, J., “Hilbert's inequalities and their reverses”, Publ. Math. Debrecen 67 (2005) 315331.
[7]Krylov, V. I., Approximate calculation of integrals (Macmillan, New York, 1962).
[8]Levin, V., “On the two parameter extension and analogue of Hilbert's inequality”, J. London Math. Soc. 11 (1936) 119124.
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A Hilbert inequality and an Euler-Maclaurin summation formula

  • Mario Krnić and Josip Pečarić

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