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In this paper we present the concept of bounded second variation of a real valued function defined on a rectangle in ${ℝ}^2$. We use Hardy-Vitali type technics in the plane in order to extend the classical notion of function of bounded second variation on intervals of $ℝ$. We introduce the class $B{V}^2\left(I_a^b\right)$, of all functions of bounded second variation on a rectangle $I_a^b\subset {ℝ}^2$, and show that this class can be equipped with a norm with respect to which it is a Banach space. Finally, we present two results that show that integrals...

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...

We present the notion of bounded second $\kappa$-variation for real functions defined on an interval $[a,b]$. We introduce the class $\kappa B{V}^2\left([a,b]\right)$ of all functions of bounded second $\kappa$-variation on $[a,b]$. We show several properties of this class and present a sufficient condition under which a composition operator acts between these spaces.