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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 ($ \Omega $, $ \Sigma $, $ \mu $) be a measure space with two sets $ A,B \in \Sigma $ such that $ 0 < \mu (A) < 1 < \mu (B) < \infty $ , and let $ \varphi :{{\mathbf{R}}_ + } \to {{\mathbf{R}}_ + }$ be bijective and $ {\varphi ^{ - 1}}$ continuous at 0. We prove that if for all $ \mu $-integrable step functions $ x,y:\Omega \to {\mathbf{R}}$,

$\displaystyle {\varphi ^{ - 1}}\left( {\int_\Omega {\varphi \circ \vert x + y\v... ...varphi ^{ - 1}}\left( {\int_\Omega {\varphi \circ \vert y\vert d\mu } } \right)$

then $ \varphi (t) = \varphi (1){t^p}$ for some $ p \geq 1$. In the case of normalized measure we prove a generalization of Minkowski's inequality theorem. The suitable results for the reversed inequality are also presented.

References [Enhancements On Off] (What's this?)

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Keywords: Measure space, Minkowski's inequality, subadditive functions, convex functions, normalized measure
Article copyright: © Copyright 1990 American Mathematical Society

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