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Some second-order integral inequalities of generalized Hardy type

Published online by Cambridge University Press:  14 November 2011

B. Florkiewicz  and
K. Wojteczek
B. Florkiewicz
Affiliation:
Institute of Mathematics, Wroclaw University of Technology, Wybrzeże Wyspiańkiego 27, 50-370 Wroclaw., Poland
K. Wojteczek
Affiliation:
Institute of Mathematics Technical University of Opole Luboszycka 3 45-036 OpolePoland (kwoj@polo.po.opole.pl)
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Abstract

A uniform method of obtaining various types of integral inequalities involving a function and its first derivative is extended to integral inequalities involving a function and its second derivative. Specifically, some quadratic integral inequalities of generalized Hardy type involving a function and its second derivative are derived and examined. The functions for which the inequalities hold are characterized by boundary conditions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 5 , 1999 , pp. 947 - 958
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Beesack, P. R.. Integral inequalities of the Wirtinger type. Duke Math. J. 25 (1958), 477–498.CrossRefGoogle Scholar
2Brown, R. C. and Hinton, D. B.. A weighted Hardy's inequality and non-oscillatory differential equations. Quaes. Mathematicae 15 (1992), 197–212.CrossRefGoogle Scholar
3Coles, W. J.. A general Wirtinger-type inequality. Duke Math. J. 27 (1960), 133–138.CrossRefGoogle Scholar
4Coles, W. J.. Wirtinger-type integral inequalities. Pacific J. Math. 11 (1961), 871–877.CrossRefGoogle Scholar
5Florkiewicz, B.. Some integral inequalities of Hardy type. Colloq. Math. 43 (1980), 321–330.CrossRefGoogle Scholar
6Florkiewicz, B. and Kuchta, M.. Some quadratic integral inequalities of first order. Colloq. Math. 75 (1998), 7–18.CrossRefGoogle Scholar
7Florkiewicz, B. and Rybarski, A.. Some integral inequalities of Sturm–Liouville type. Colloq. Math. 36 (1976), 127–141.CrossRefGoogle Scholar
8Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities, 2nd edn (Cambridge University Press, 1991).Google Scholar
9Kuchta, M.. Some quadratic integral inequalities of Opial type. Ann. Polon. Math. 63 (1996), 103–113.CrossRefGoogle Scholar
10Kufner, A.. Higher-order Hardy inequalities. Bayreuth. Math. Schrift. 44 (1993), 105–146.Google Scholar
11Kufner, A. and Sinnamon, G.. Overdetermined Hardy inequalities. J. Math. Anal. Appl. 213 (1997), 468–486.CrossRefGoogle Scholar
12Kufner, A. and Wannebo, A.. Some remarks on the Hardy inequality for higher-order derivatives, General Inequalities 6. International Series of Numerical Mathematics (ed. Walter, W.), vol. 103, pp. 33–48 (Basel: Birkhaüser, 1992).Google Scholar
13Martin-Reyes, F. J. and Sawyer, E.. Weighted inequalities for Riemann–Liouville fractional integrals of order one and greater. Proc. Am. Math. Soc. 106 (1989), 727–733.CrossRefGoogle Scholar
14Mitrinović, D. S., Pečarić, J. E. and Fink, A. M.. Inequalities involving functions and their integrals and derivatives (Dordrecht: Kluwer Academic, 1991).CrossRefGoogle Scholar
15Opic, B. and Kufner, A.. Hardy-type inequalities. Pitman Research Notes in Mathematics 219 (Harlow: Longman Scientific and Technical, 1990).Google Scholar
16Sinnamon, G. J.. Weighted Hardy and Opial-type inequalities. J. Math. Analysis Applic. 160 (1991), 434–445.CrossRefGoogle Scholar
17Stepanov, V.. Two weighted estimates for Riemann–Liouville integrals. Math. USSR Izvestiya 36 (1991), 669–681.CrossRefGoogle Scholar
18Swanson, C. A.. Picone's identity. Rend. Matematica 8 (1975), 373–397.Google Scholar