On Korenblum convex functions. (English) Zbl 1416.26022

Summary: We introduce a new class of generalized convex functions called the \(\kappa\)-convex functions, based on Korenblum’s concept of \(\kappa\)-decreasing functions, where \(\kappa\) 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 \(\kappa\)-variation, extending a result of F. Riesz. We also present a formal structural decomposition result for the \(\kappa\)-convex functions.


26A51 Convexity of real functions in one variable, generalizations
39B62 Functional inequalities, including subadditivity, convexity, etc.
26B25 Convexity of real functions of several variables, generalizations
26A48 Monotonic functions, generalizations
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