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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.

MSC:

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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