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Generalizations of Hardy's Integral Inequality

Published online by Cambridge University Press:  14 November 2011

E. R. Love
E. R. Love
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
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Synopsis

Extensions of the integral version of Hardy's Inequality were given by Kadlec and Kufner (1967) and by Copson (1976). This paper provides several levels of further generalization of their results, obtained mostly by specializing four main inequalities. Most of the inequalities have the form ∥Kf∥ ≤ C ∥f∥, where K is an integral transform and ∥.∥ is a generalized Lp-norm; some have the inequality sign reversed. Best possible constants C are obtained in several cases, under mild extra hypotheses.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 3-4 , 1985 , pp. 237 - 262
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Hardy, G. H., Littlewood, J. E. and Pölya, G.. Inequalities (Cambridge University Press, 1934).Google Scholar
2Copson, E. T.. Some integral inequalities. Proc. Roy. Soc. Edinburgh Sect. A 75 (1975–1976), 157–164.CrossRefGoogle Scholar
3Kadlec, J. and Kufner, A.. Characterization of functions with zero traces by integrals with weight functions II. Časopis Pêst. Mat. 92 (1967), 16–28.CrossRefGoogle Scholar