Counter-examples to Breckner-convexity

Házy, Attila and Makó, Judit (2023) Counter-examples to Breckner-convexity. MULTIDISZCIPLINÁRIS TUDOMÁNYOK: A MISKOLCI EGYETEM KÖZLEMÉNYE, 13 (3). pp. 74-80. ISSN 2062-9737

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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.

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