COUNTER-EXAMPLES CONCERNING BRECKNER-CONVEXITY

In this paper, we examine convexity type inequalities. Let D be a nonempty convex subset of a linear space, c > 0 and alpha : D-D -> R be a given even function. The inequality f (x+y/2) <= cf(x) +cf(y) + alpha(x-y) (x,y is an element of D) is the focus of our examinations. We will construct an example to show that for c = 1, this Jensen type inequality does not imply the convexity of the function. Then, we compare this inequality with Hermite-Hadamard type inequalities.