Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T01:09:33.243Z Has data issue: false hasContentIssue false
  • English
  • Français

A Hilbert inequality and an Euler-Maclaurin summation formula

Published online by Cambridge University Press:  17 February 2009

Mario Krnić  and
Josip Pečarić
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

inequalitiesHilbert's inequalityBernoulli numbersBernoulli polynomialsEuler-Maclaurin summation formulahomogeneous functionkernelweight functionequivalent inequalities.
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 3 , January 2007 , pp. 419 - 431
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Bicheng, Y., “On an extension of Hardy-Hilbert's inequality with some parameters”, Math. Inequal. Appl. (to appear).Google Scholar
[2]Bonsall, F. F., “Inequalities with non-conjugate parameters”, Quart. J. Math. Oxford (2) 2 (1951) 135–150.CrossRefGoogle Scholar
[3]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge Univ. Press, Cambridge, 1952).Google Scholar
[4]Jichang, K. and Debnath, L., “On new generalizations of Hilbert's inequality and their applications”, Math. Inequal. Appl. 245 (2000) 248–265.Google Scholar
[5]Krnić, M. and Pečarić, J., “General Hilbert's and Hardy's inequalities”, Math. Inequal. Appl. 8 (2005) 29–51.Google Scholar
[6]Krnić, M. and Pečarić, J., “Hilbert's inequalities and their reverses”, Publ. Math. Debrecen 67 (2005) 315–331.CrossRefGoogle Scholar
[7]Krylov, V. I., Approximate calculation of integrals (Macmillan, New York, 1962).Google Scholar
[8]Levin, V., “On the two parameter extension and analogue of Hilbert's inequality”, J. London Math. Soc. 11 (1936) 119–124.CrossRefGoogle Scholar