Skip to main content Accessibility help
×
Home
Hostname: page-component-544b6db54f-d2wc8 Total loading time: 0.167 Render date: 2021-10-17T04:41:42.401Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES VIA CONVEXITY

Published online by Cambridge University Press:  01 April 2008

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
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 AB. 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.

Type
Research Article
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.

References

[1]Beesack, P. R. and Heinig, H. P., ‘Hardy’s inequalities with indices less than 1’, Proc. Amer. Math. Soc. 83(3) (1981), 532536.Google Scholar
[2]Čižmešija, A. and Pečarić, J., ‘Some new generalizations of inequalities of Hardy and Levin–Cochran–Lee’, Bull. Austral. Math. Soc. 63(1) (2001), 105113.Google Scholar
[3]Čižmešija, A. and Pečarić, J., ‘Multivariable mixed means and inequalities of Hardy and Levin–Cochran–Lee type’, Math. Inequal. Appl. 5(3) (2002), 397415.Google Scholar
[4]Čižmešija, A., Pečarić, J. and Persson, L.-E., ‘On strengthened Hardy and Pólya–Knopp’s inequalities’, J. Approx. Theory 125 (2003), 7484.CrossRefGoogle Scholar
[5]Hardy, G. H., ‘Note on a theorem of Hilbert’, Math. Z. 6 (1920), 314317.CrossRefGoogle Scholar
[6]Hardy, G. H., ‘Notes on some points in the integral calculus LX: An inequality between integrals’, Messenger of Math. 54 (1925), 150156.Google Scholar
[7]Hardy, G. H., ‘Notes on some points in the integral calculus LXIV’, Messenger of Math. 57 (1928), 1216.Google Scholar
[8]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge University Press, Cambridge, 1959).Google Scholar
[9] H. P. Heinig, ‘Variations of Hardy’s inequality’, Real Anal. Exchange 5 (1979–80), 61–81.Google Scholar
[10]Kaijser, S., Nikolova, L., Persson, L.-E. and Wedestig, A., ‘Hardy type inequalities via convexity’, Math. Inequal. Appl. 8(3) (2005), 403417.Google Scholar
[11]Kaijser, S., Persson, L.-E. and Öberg, A., ‘On Carleman and Knopp’s inequalities’, J. Approx. Theory 117 (2002), 140151.CrossRefGoogle Scholar
[12]Knopp, K., ‘Über Reihen mit positiven Gliedern’, J. London Math. Soc. 3 (1928), 205211.CrossRefGoogle Scholar
[13]Kufner, A., Maligranda, L. and Persson, L.-E., ‘The prehistory of the Hardy inequality’, Amer. Math. Monthly 113 (2006), 715732.CrossRefGoogle Scholar
[14]Kufner, A., Maligranda, L. and Persson, L.-E., The Hardy inequality—about its history and some related results (Vydavatelsky Servis Publishing House, Pilsen, 2007).Google Scholar
[15]Kufner, A. and Persson, L.-E., Weighted inequalities of Hardy type (World Scientific, Singapore, 2003).CrossRefGoogle Scholar
[16]Oguntuase, J. A., Okpoti, C. A., Persson, L.-E. and Allotey, F. K. A., ‘Multidimensional Hardy type inequalities for p<0 and 0<p<1’, J. Math. Inequal. 1 (2007), 111.CrossRefGoogle Scholar
[17]Oguntuase, J. A., Okpoti, C. A., Persson, L.-E. and Allotey, F. K. A., ‘Weighted multidimensional Hardy and Pólya–Knopp’s type inequalities’, Research report 15, Department of Mathematics, Luleå University of Technology, Sweden, 2006, 15 pages.Google Scholar
[18]Opic, B. and Kufner, A., Hardy-type inequalities, Pitman Series, 219 (Longman Scientific & Technical, Harlow, 1990).Google Scholar
[19]Pachpatte, B. G., ‘On multivariate Hardy type inequalities’, An. Stiint. Al. I. Cuza Iasi Sect. I a Mat. 38(3) (1992), 355361.Google Scholar
[20]Pachpatte, B. G., ‘On some generalizations of Hardy’s integral inequality for functions of several variables’, Demonstratio Math. 27(1) (1994), 4351.Google Scholar
You have Access
5
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES VIA CONVEXITY
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES VIA CONVEXITY
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES VIA CONVEXITY
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *