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On Integral Inequalities Related to Hardy's

Published online by Cambridge University Press:  20 November 2018

D. T. Shum
D. T. Shum*
Affiliation:
Carleton University, Ottawa, Ontario
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The purpose of this note is to provide integral inequalities which are related to Hardy's ([2] and [3, Theorem 330]). This latter result we state as

Theorem 1. Let p>1, r≠1, and ƒ(x) be nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1. If F(x) is defined by

1

then

2

unless f≡0. The constant is the best possible.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 2 , June 1971 , pp. 225 - 230
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Benson, D. C., Inequalities involving integrals of functions and their derivatives, J. Math. Anal. Appl. 17 (1967), 292-308.Google Scholar
2. Hardy, G. H., Note on some points in the integral calculus (LXIV), Messenger of Math. 57 (1928), 12-16.Google Scholar
3. Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities, second edition, Cambridge, 1952.Google Scholar