On Approximately h-Convex Functions

A real valued function

defined on an open convex subset

D D

of a normed space

X X

is called rationally

(h,d) ( 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) f ( t x + ( 1 − t ) y ) ≤ h ( t ) f ( x ) + h ( 1 − t ) f ( y ) + d ( x , y )

for all

x,y\in D x , y ∈ D

and

t\in {\mathbb Q}\cap [0,1] t ∈ Q ∩ [ 0 , 1 ]

, where

d\colon X \times X \to {\mathbb R} d  ⁣ : X × X → R

and

h:[0,1] \to {\mathbb R} h : [ 0 , 1 ] → R

are given functions. Our main result is of Bernstein-Doetsch type. Namely, we prove that if

f f

is locally bounded from above at a point of

D D

and rationally

(h,d) ( h , d )

-convex then it is continuous and

(h,d) ( h , d )

-convex.