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Journal of Convex Analysis 18 (2011), No. 2, 447--454
Copyright Heldermann Verlag 2011

On Approximately h-Convex Functions

Pŕl Burai
Dept. of Applied Mathematics and Probability Theory, University of Debrecen, 4010 Debrecen Pf. 12, Hungary

Attila Házy
Dept. of Applied Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary


\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 $$ f\left(tx + (1-t)y \right) \leq h(t) f(x) + h(1-t) f(y) + d(x,y) $$ 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 Bernstein-Doetsch 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, h-convexity, s-convexity, Bernstein-Doetsch theorem, regularity properties of generalized convex functions.

MSC: 26A51, 26B25, 39B62

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