Counter-examples to Breckner-convexity

Abstract

In this paper, we examine convexity type inequalities. Let D be a nonempty convex subset of a linear space, c>0 and α:D-D→R be a given even function. The inequality

                                        f((x+y)/2) ≤ c f(x) + c f(y) + α(x-y)  (x,y ∈ 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.