Abstract. In this paper we collect some properties of strongly midconvex functions. First, counterparts of the classical theorems of Bernstein-Doetsch, Ostrowski and Sierpiński are presented. A version of Rodé support theorem for strongly midconvex functions and a Kuhn-type result on the relation between strongly midconvex functions and strongly t-convex functions are obtained. Finally, a connection between strong midconvexity and generalized convexity in the sense of Beckenbach is established.

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Abstract. We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function.

The purpose of this paper is twofold. Firstly, we introduce the concept of boundedκΦ-variation in the sense of Schramm-Korenblum for real functions with domain in a rectangle ofR2. Secondly, we study some properties of these functions and we prove that the space generated by these functions has a structure of Banach algebra.