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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A functional inequality and its relation to convexity of vector-valued functions

Author: Ih Ching Hsu
Journal: Proc. Amer. Math. Soc. 58 (1976), 119-123
MSC: Primary 46A40; Secondary 26A51
MathSciNet review: 0415265
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Abstract: With respect to a partial ordering $ \ll $, the functional inequality $ F(s) + tG(s) \ll F(s + t)$ arises naturally in the study of extending classical convex-function theory to vector-valued functions. The solution $ F$ is strongly convex and has a Riemann type integral representation, even a Bochner type integral representation when the functional inequality is considered in a Banach lattice. The paper also proves the equivalence of strong and weak convexity in an ordered locally convex space whose positive cone is closed. As an application, an affirmative answer is given to an open question raised earlier by R. G. Kuller and the author.

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Keywords: Weak convexity, strong convexity, Hahn-Banach Theorem, ordered locally convex space, positive cone, order-convergence, Banach lattice, order-bounded, strong Lebesgue measurable, Bochner integral, Riesz descomposition theorem
Article copyright: © Copyright 1976 American Mathematical Society

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