On Approximately Midconvex Functions

A real-valued function f defined on an open, convex set D ofa real normed space is called (, )-midconvex if it satisfies The main result of the paper states that if f is locally boundedfrom above at a point of D and is (, )-midconvex, then it satisfiesthe convexity-type inequality where : 0, 1] R is a continuous function satisfying The particular case = 0 of this result is due to Ng and Nikodem(Proc. Amer. Math. Soc. 118 (1993) 103–108), while thespecialization = = 0 yields the theorem of Bernstein and Doetsch(Math. Ann. 76 (1915) 514–526). 2000 Mathematics SubjectClassification 26A51, 26B25.