Subadditive functions and a relaxation of the homogeneity condition of seminorms
Author:
Janusz Matkowski
Journal:
Proc. Amer. Math. Soc. 117 (1993), 9911001
MSC:
Primary 26A12; Secondary 39B72, 46B99
MathSciNet review:
1113646
Fulltext PDF Free Access
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Abstract: We prove that every locally bounded above at a point subadditive function such that , for some has to be linear. Using this we show among others that the homogeneity condition of a seminorm in a real linear space can be essentially relaxed to the following condition: there exists an such that for all . A new characterization of the norm and oneline proofs of Minkowski's and Höder's inequalities are also given.
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 [1]
 E. Berz, Sublinear functions on , Aequationes Math. 12 (1975), 200206. MR 0387862 (52:8700)
 [2]
 A. Bruckner, Minimal superadditive extensions of superadditive functions, Pacific J. Math. 10 (1960), 11551162. MR 0122943 (23:A275)
 [3]
 E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, RI, 1957. MR 0089373 (19:664d)
 [4]
 M. Kuczma, An introduction to the theory of functional equations and inequalities, Prace Nauk. Uniw. Śląsk. Katowice, vol. 489, Polish Scientific Publ., Warsaw, Krakow, and Katowice, 1985. MR 788497 (86i:39008)
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 J. Matkowski, On a characterization of norm, Ann. Polon. Math. 50 (1989), 137144. MR 1044861 (91c:46042)
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 , Functional inequality characterizing convex functions, conjugacy and a generalization of Hölder's and Minkowski's inequalities, Aequationes Math. 40 (1990), 168180. MR 1069792 (91i:39005)
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 J. Matkowski and T. Swiątkowski, Quasimonotonicity, subadditive bijections of and characterization of norm, J. Math. Anal. Appl. 154 (1991), 493506. MR 1088646 (92g:26014)
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 J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663675, MR 1009994 (90m:39025)
 [9]
 , A generalization of Mulholland inequality (submitted).
 [10]
 H. P. Mulholland, On generalizations of Minkowski's inequality in the form of a triangle inequality, Proc. London Math. Soc. (2) 51 (1950), 294307. MR 0033865 (11:503f)
 [11]
 R. A. Rosenbaum, Subadditive functions, Duke Math. J. 17 (1950), 227242. MR 0036796 (12:164a)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199311136469
PII:
S 00029939(1993)11136469
Keywords:
Subadditive functions,
seminorm,
measure space,
characterization of norm,
Minkowski's inequality,
Hölder's inequality
Article copyright:
© Copyright 1993
American Mathematical Society
