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Subadditive functions and a relaxation of the homogeneity condition of seminorms

Author: Janusz Matkowski
Journal: Proc. Amer. Math. Soc. 117 (1993), 991-1001
MSC: Primary 26A12; Secondary 39B72, 46B99
MathSciNet review: 1113646
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Abstract: We prove that every locally bounded above at a point subadditive function $ f:(0,\infty ) \to \mathbb{R}$ such that $ f(rt) \leqslant rf(t),\;t > 0$, for some $ r \in (0,1)$ has to be linear. Using this we show among others that the homogeneity condition of a seminorm $ {\mathbf{p}}$ in a real linear space $ X$ can be essentially relaxed to the following condition: there exists an $ r \in (0,1)$ such that $ {\mathbf{p}}(rx) \leqslant r{\mathbf{p}}(x)$ for all $ x \in X$. A new characterization of the $ {{\mathbf{L}}^p}$-norm and one-line proofs of Minkowski's and Höder's inequalities are also given.

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Keywords: Subadditive functions, seminorm, measure space, characterization of $ {{\mathbf{L}}^p}$-norm, Minkowski's inequality, Hölder's inequality
Article copyright: © Copyright 1993 American Mathematical Society

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