On Korenblum convex functions

We introduce a new class of generalized convex functions called the κ-convex functions, based on Korenblum's concept of κ-decreasing functions, where κ is an entropy (distortion) function. We study continuity and differentiability properties of these functions, and we discuss a special subclass which is a counterpart of the class of so-called d.c. functions. We characterize this subclass in terms of the space of functions of bounded second κ-variation, extending a result of F. Riesz. We also present a formal structural decomposition result for the κ-convex functions.