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On Mulholland's inequality

Author(s): Witold Jarczyk; Janusz Matkowski
Journal: Proc. Amer. Math. Soc. 130 (2002), 3243-3247.
MSC (2000): Primary 39B72, 26D15; Secondary 26A51
Posted: June 11, 2002
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Abstract: H.P. Mulholland has presented a sufficient condition for a generalization of the Minkowski inequality and another such condition was given by R.M. Tardiff. We show that Mulholland's condition implies Tardiff's, but that the converse is false.

References:

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M. Kuczma, An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality, Uniwersytet Slaski, Polish Scientific Publishers, Warszawa-Kraków-Katowice 1985. MR 86i:39008

[2]
J. Matkowski, $L^p$-like paranorms, in: Selected Topics in Functional Equations and Iteration Theory, Proceedings of the Austrian-Polish Seminar, Graz, 1991, Grazer Math. Ber. 316 (1992), 103-138. MR 94g:39010

[3]
J. Matkowski, On a generalization of Mulholland's inequality, Abh. Mat. Sem. Univ. Hamburg 63 (1993), 97-103. MR 94g:26032

[4]
H.P. Mulholland, On generalizations of Minkowski's inequality in the form of a triangle inequality, Proc. London Math. Soc. (2) 51 (1950), 294-307. MR 11:503f

[5]
A. Sklar, Remark and Problem, Report of meeting, 37th International Symposium on Functional Equations (Huntington, 1999), Aequationes Math. 60 (2000), 187-188.

[6]
R. M. Tardiff, On a generalized Minkowski inequality and its relation to dominates for $t$-norms, Aequationes Math. 27 (1984), 308-316. MR 86j:26013

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Additional Information:

Witold Jarczyk
Affiliation: Institute of Mathematics, Technical University, Podgórna 50, PL--65--246 Zielona Góra, Poland -- and -- Institute of Mathematics, Silesian University, Bankowa 14, PL--40--007 Katowice, Poland
Email: w.jarczyk@im.uz.zgora.pl

Janusz Matkowski
Affiliation: Institute of Mathematics, Technical University, Podgórna 50, PL--65--246 Zielona Góra, Poland -- and -- Institute of Mathematics, Silesian University, Bankowa 14, PL--40--007 Katowice, Poland
Email: j.matkowski@im.uz.zgora.pl

DOI: 10.1090/S0002-9939-02-06708-4
PII: S 0002-9939(02)06708-4
Keywords: Generalizations of the Minkowski inequality, Mulholland's inequality, convex functions, geometrically convex functions
Received by editor(s): June 4, 2001
Posted: June 11, 2002
Communicated by: Jonathan M. Borwein
Copyright of article: Copyright 2002, American Mathematical Society

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