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On Korenblum convex functions

Ereú J., López L., Merentes N.

Commentationes Mathematicae

2017

Vol 57, No. 1

25--44

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

Uniformly continuous superposition operators in the space of functions of bounded n-dimensional Φ-variation

Bracamonte M., Ereú J., Giménez J., Merentes N.

Demonstratio Mathematica

2014

Vol. 47, nr 1

56--68

We prove that if a superposition operator maps a subset of the space of all metric-vector-space-valued-functions of bounded n-dimensional Φ-variation into another such space, and is uniformly continuous, then the generating function of the operator is an affine function in the functional variable.

On bi-dimensional second μ-variation

Ereú J., Giménez J., Merentes N.

Demonstratio Mathematica

2014

Vol. 47, nr 4

910--932

In this paper, we present a generalization of the notion of bounded slope variation for functions defined on a rectangle Iba in R2. Given a strictly increasing function μ, defined in a closed real interval, we introduce the class BVμ,2 (Iba), of functions of bounded second μ-variation on Iba ; and show that this class can be equipped with a norm with respect to which it is a Banach space. We also deal with the important case of factorizable functions in BVμ,2 (Iba) and finally we exhibit a relation between this class and the one of double Riemann–Stieltjes integrals of functions of bi-dimensional bounded variation.