The converse of the Minkowski's inequality theorem and its generalization

Author:
Janusz Matkowski

Journal:
Proc. Amer. Math. Soc. **109** (1990), 663-675

MSC:
Primary 39C05; Secondary 26D10, 26D15, 46E30

MathSciNet review:
1009994

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Abstract: Let (, , ) be a measure space with two sets such that , and let be bijective and continuous at 0. We prove that if for all -integrable step functions ,

**[1]**J. Aczél,*Lectures on functional equations and their applications*, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York-London, 1966. Translated by Scripta Technica, Inc. Supplemented by the author. Edited by Hansjorg Oser. MR**0208210****[2]**G. H. Hardy, J. E. Littlewood, and G. Pólya,*Inequalities*, Cambridge, at the University Press, 1952. 2d ed. MR**0046395****[3]**Marek Kuczma,*An introduction to the theory of functional equations and inequalities*, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 489, Uniwersytet Śląski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality; With a Polish summary. MR**788497****[4]**Norbert Kuhn,*A note on 𝑡-convex functions*, General inequalities, 4 (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 71, Birkhäuser, Basel, 1984, pp. 269–276. MR**821804****[5]**J. Matkowski and T. Swiatkowski,*Quasi-monotontcity, subadditive bijections of**and characterization of**-norm*, J. Math. Anal, and Appl. (to appear).**[6]**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**0033865**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1009994-0

Keywords:
Measure space,
Minkowski's inequality,
subadditive functions,
convex functions,
normalized measure

Article copyright:
© Copyright 1990
American Mathematical Society