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BanachJ.Math.Anal.8(2014),no.1,1–13 ONTHEEQUIVALENCEBETWEENSOMEMULTIDIMENSIONALHARDY-TYPEINEQUALITIES B J M A

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Banach J. Math. Anal. 8 (2014), no. 1, 1–13

B

anach

J

ournal of

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athematical

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nalysis ISSN: 1735-8787 (electronic)

www.emis.de/journals/BJMA/

ON THE EQUIVALENCE BETWEEN SOME MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES

J. A. OGUNTUASE1∗, L.-E. PERSSON2, N. SAMKO2 AND A. SONUBI1 Communicated by M. A. Ragusa

Abstract. We prove and discuss some power weighted Hardy-type inequali- ties on finite and infinite sets. In particular, it is proved that these inequalities are equivalent because they can all be reduced to an elementary inequality, which can be proved by Jensen inequality. Moreover, the corresponding limit (P´olya–Knopp type) inequalities and equivalence theorem are proved. All con- stants in these inequalities are sharp.

1 Department of Mathematics, Federal University of Agriculture, P.M.B.

2240, Abeokuta, Ogun State, Nigeria.

E-mail address: [email protected] E-mail address: [email protected]

2 Department of Mathematics, Lule˚a University of Technology, SE-971 87 Lule˚a, Sweden; Narvik University College, P.O. Box 385, N-8505 Narvik, Norway.

E-mail address: [email protected] E-mail address: [email protected]

Date: Received: 7 November 2012; Accepted: 24 February 2013.

∗ Corresponding author.

2010Mathematics Subject Classification. Primary 26D15; Secondary 26A51, 47J20.

Key words and phrases. Inequality, multidimensional Hardy-type inequalities, multidimen- sional P´olya–Knopp type inequalities, best constant, power weights, convexity.

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