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 , the functional inequality arises naturally in the study of extending classical convex-function theory to vector-valued functions. The solution 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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DOI:
https://doi.org/10.1090/S0002-9939-1976-0415265-6

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