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Inequalities for a function involving its integral and derivative

Published online by Cambridge University Press:  14 November 2011

Horng Jaan Li  and
Cheh Chih Yeh
Horng Jaan Li
Affiliation:
Department of Mathematics, National Central University, Chung-Li, Taiwan 32054
Cheh Chih Yeh
Affiliation:
Department of Mathematics, National Central University, Chung-Li, Taiwan 32054
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Abstract

We give a concise approach to generalising the inequalities of Wirtinger, Hardy, Weyl and Opial by using the well-known inequality: if X and Y are non-negative, then

for p > 1 (0 < p < 1), respectively.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 1 , 1995 , pp. 133 - 151
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Beesack, P. R.. On some integral inequalities of E. T. Copson. In General Inequalities 2, ed. Beckenbach, E. F. (Basel: Birkhauser, 1980).Google Scholar
2Beesack, P. R.. Hardy's inequality and its extensions. Pacific J. Math. 11 (1961), 39–61.CrossRefGoogle Scholar
3Beescak, P. R.. On an integral inequality of Z. Opial. Trans. Amer. Math. Soc. 104 (1962), 470–475.CrossRefGoogle Scholar
4Beesack, P. R.. Integral inequalities involving a function and its derivative. Amer. Math. Monthly 78 (1971), 705–741.CrossRefGoogle Scholar
5Benson, D. C.. Inequalities involving integrals of functions and their derivatives. J. Math. Anal. Appl. 17 (1967), 292–308.CrossRefGoogle Scholar
6Boyd, D. W.. Best constants in a class of integral inequalities. Pacific J. Math. 30 (1969), 367–383.CrossRefGoogle Scholar
7Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities, 2nd edn (Cambridge: Cambridge University Press, 1988).Google Scholar
8Mitrinovic, D. S., Pecaric, J. E. and Fink, A. M.. Inequalities Involving Functions and Their Integrals and Derivatives (Dordrecht: Kluwer, 1991).CrossRefGoogle Scholar
9Shum, D. T.. On integral inequalities related to Hardy's. Canad. Math. Bull. 14 (1971), 225–230.CrossRefGoogle Scholar
10Yang, G. S.. On a certain result of Z. Opial. Proc. Japan Acad. 42 (1966), 78–83.Google Scholar