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

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.