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MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES VIA CONVEXITY

Part of: Inequalities

Published online by Cambridge University Press:  01 April 2008

JAMES A. OGUNTUASE ,
LARS-ERIK PERSSON  and
ALEKSANDRA ČIŽMEŠIJA
JAMES A. OGUNTUASE
Affiliation:
Department of Mathematics, University of Agriculture, P M B 2240, Abeokuta, Nigeria (email: oguntuase@yahoo.com)
LARS-ERIK PERSSON
Affiliation:
Department of Mathematics, Luleå University of Technology, SE – 971 87, Luleå, Sweden (email: larserik@sm.luth.se)
ALEKSANDRA ČIŽMEŠIJA*
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia (email: cizmesij@math.hr)
*
✉For correspondence; e-mail: cizmesij@math.hr
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Abstract

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Let an almost everywhere positive function Φ be convex for p>1 and p<0, concave for p∈(0,1), and such that Axp≤Φ(x)≤Bxp holds on for some positive constants A≤B. In this paper we derive a class of general integral multidimensional Hardy-type inequalities with power weights, whose left-hand sides involve instead of , while the corresponding right-hand sides remain as in the classical Hardy’s inequality and have explicit constants in front of integrals. We also prove the related dual inequalities. The relations obtained are new even for the one-dimensional case and they unify and extend several inequalities of Hardy type known in the literature.

Keywords

inequalitiesintegral inequalitiesmultidimensional inequalitiesHardy’s inequalityweightspower weightsJensen’s inequalityconvex functions

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 2 , April 2008 , pp. 245 - 260
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The research of the first author was supported by the Swedish Institute under the Guest Fellowship Programme 210/05529/2005. The research of the third author was supported by the Croatian Ministry of Science, Education and Sports, under Research Grant 058-1170889-1050.

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