
Journal of Convex Analysis 18 (2011), No. 2, 447454 Copyright Heldermann Verlag 2011 On Approximately hConvex Functions Pŕl Burai Dept. of Applied Mathematics and Probability Theory, University of Debrecen, 4010 Debrecen Pf. 12, Hungary burai@inf.unideb.hu Attila Házy Dept. of Applied Mathematics, University of Miskolc, 3515 MiskolcEgyetemváros, Hungary matha@unimiskolc.hu [Abstractpdf] \def\R{\mathbb R} \def\Q{\mathbb Q} A real valued function $f\colon D\to \R$ defined on an open convex subset $D$ of a normed space $X$ is called \emph{rationally $(h,d)$convex} if it satisfies $$ f\left(tx + (1t)y \right) \leq h(t) f(x) + h(1t) f(y) + d(x,y) $$ for all $x,y\in D$ and $t\in \Q \cap [0,1]$, where $d\colon X \times X \to \R$ and $h:[0,1] \to \R$ are given functions. \par Our main result is of BernsteinDoetsch type. Namely, we prove that if $f$ is locally bounded from above at a point of $D$ and rationally $(h,d)$convex then it is continuous and $(h,d)$convex. Keywords: Convexity, approximate convexity, hconvexity, sconvexity, BernsteinDoetsch theorem, regularity properties of generalized convex functions. MSC: 26A51, 26B25, 39B62 [ Fulltextpdf (108 KB)] for subscribers only. 