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Hardy-type inequalities for means

Published online by Cambridge University Press:  17 April 2009

Zsolt Páles  and
Lars-Erik Persson
Zsolt Páles
Affiliation:
Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, pf.12, Hungary, e-mail: pales@math.klte.hu
Lars-Erik Persson
Affiliation:
Department of Mathematics, LuleåUniversity of Technology, Luleå, Sweden, e-mail: larserik@sm.luth.se
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In this paper we consider inequalities of the form , Where M is a mean. The main results of the paper offer sufficient conditions on M so that the above inequality holds with a finite constant C. The results obtained extend Hardy's and Carleman's classical inequalities together with their various generalisations in a new dirction.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 3 , December 2004 , pp. 521 - 528
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Aczél, J. and Daróczy, Z., ‘Über verallgemeinerte quasilineare Mittelwerte, die mit Gewichtsfunktionen gebildet sind’, Publ. Math. Debrecen 10 (1963), 171–190.Google Scholar
[2]Bajraktarević, M., ‘Sur une équation fonctionelle aux valeurs moyennes’, Glasnik Mat.–Fiz. Astronom. Društvo Mat. Fiz. Hrvatske. Ser. II 13 (1958), 243–248.Google Scholar
[3]Bajraktarević, M., ‘Uber die Vergleichbarkeit der mit Gewichtsfunktionen gebildten Mittelwerte’, Studia Sci. Math. Hungar. 4 (1969), 3–8Google Scholar
[4]Daróczy, Z., ‘Über eine Klasse von Mittelwerten’, Publ. Math. Debrecen 19 (1972), 211–217.CrossRefGoogle Scholar
[5]Daróczy, Z. and Losonczi, L., ‘Über den Vergleich von Mittelwerten’, Publ. Math. Debrecen 17 (1970), 289–297.CrossRefGoogle Scholar
[6]Gini, C., ‘Di una formula compressiva delle media’, Metron 13 (1938), 3–22.Google Scholar
[7]Hardy, G.H., ‘Notes on some points in the integral calculus, LI’ Messenger of Math. 48 (919), 107–112.Google Scholar
[8]Hardy, G.H., ‘Note on a theorem of Hilbert’, Math. Z. 6 (1920), 314–317.CrossRefGoogle Scholar
[9]Hardy, G.H., ‘Notes on some points in the integral calculus, LX’, Messenger of Math. 54 (1925), 150–156.Google Scholar
[10]Hardy, G.H., Littlewood, J.E. and Pólya, G., Inequalities (Cambridge University Press, Cambridge, 1952).Google Scholar
[11]Knopp, K., ‘Über Reihen mit positiven Gliedern’, J. London Math. Soc. 3 (1928), 205–211.CrossRefGoogle Scholar
[12]Kufner, A. and Persson, L.-E., Weighted inequalities of Hardy type (World Scientific, New Jersey, London, Singapore, Hong Kong, 2003).Google Scholar
[13]Kufner, A. and Persson, L.-E., The Hardy inequality – About its history and current status, Research Report 2002–06 (Department of Mathematics, Luleå University of Technology, 2002).Google Scholar
[14]Mitrinović, D.S., Pečarić, J., and Fink, A.M., Inequalities involving functions and their integrals and derivatives (Kluwer Acad. Publ., Dordrecht, 1991).CrossRefGoogle Scholar
[15]Mulholland, P., ‘On the generalization of Hardy's inequality’, J. London Math. Soc. 7 (1932), 208–214.Google Scholar
[16]Opic, B. and Kufner, A., Hardy–type inequalities, Pitman Research Notes in Mathematics (Longman Scientific & Technical, Harlow, New York, 1990).Google Scholar
[17]Páles, Zs., ‘Characterization of quasideviation means’, Acta Math. Acad. Sci. Hungar. 40 (1982), 243–260.Google Scholar
[18]Páles, Zs., ‘Essential inequalities for means’, Period. Math. Hungar. 21 (1990), 9–16.CrossRefGoogle Scholar
[19]Páles, Zs., ‘Nonconvex functions and separation by power means’, Math. Ineqal. Appl. 3 (2000), 169–176.Google Scholar
[20]Pečarić, J. and Stolarsky, K.B., ‘Carleman's inequality: history and new generalizations’, Aequations Math. 61 (2001), 49–62.Google Scholar