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ON SOME REFINEMENTS AND CONVERSES OF MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES

Part of: Inequalities

Published online by Cambridge University Press:  06 March 2012

MARIO KRNIĆ
MARIO KRNIĆ*
Affiliation:
Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia (email: mario.krnic@fer.hr)
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Abstract

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The main objective of this paper is a study of some new refinements and converses of multidimensional Hilbert-type inequalities with nonconjugate exponents. Such extensions are deduced with the help of some remarkable improvements of the well-known Hölder inequality. First, we obtain refinements and converses of the general multidimensional Hilbert-type inequality in both quotient and difference form. We then apply the results to homogeneous kernels with negative degree of homogeneity. Finally, we consider some particular settings with homogeneous kernels and weighted functions, and compare our results with those in the literature.

Keywords

Hilbert-type inequalitynonconjugate exponentsHölder’s inequalityrefinementconverse

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 3 , June 2012 , pp. 380 - 394
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Bicheng, Y. and Rassias, T. M., ‘On the way of weight coefficients and research for the Hilbert-type inequalities’, Math. Inequal. Appl. 6 (2003), 625–658.Google Scholar
[2]Bonsall, F. F., ‘Inequalities with nonconjugate parameters’, Quart. J. Math. Oxford Ser. 2(2) (1951), 135–150.CrossRefGoogle Scholar
[3]Brnetić, I. and Pečarić, J., ‘Generalization of Hilbert’s integral inequality’, Math. Inequal. Appl. 7 (2004), 199–205.Google Scholar
[4]Brnetić, I. and Pečarić, J., ‘Generalization of inequalities of Hardy-Hilbert’s type’, Math. Inequal. Appl. 7 (2004), 217–225.Google Scholar
[5]Brnetić, I., Krnić, M. and Pečarić, J., ‘Multiple Hilbert and Hardy-Hilbert inequalities with nonconjugate parameters’, Bull. Aust. Math. Soc. 71 (2005), 447–457.CrossRefGoogle Scholar
[6]Krnić, M. and Pečarić, J., ‘Hilbert’s inequalities and their reverses’, Publ. Math. Debrecen 67 (2005), 315–331.Google Scholar
[7]Li, Y. and He, B., ‘On inequalities of Hilbert’s type’, Bull. Aust. Math. Soc. 76 (2007), 1–13.Google Scholar