Generalized approximate midconvexity

Abstract

Let X be a normed space and V ⊂ X a convex set with nonempty interior. Let α: [0, ∞) → [0, ∞) be a given nondecreasing function. A function f: V → R is α(·)-midconvex if x + y f ≤ 2 f(x) + f(y) 2 + α(‖x − y‖) for x, y ∈ V. In this paper we study α(·)-midconvex functions. Using a version of Bernstein-Doetsch theorem we prove that if f is α(·)-midconvex and locally bounded from above at every point then f(rx + (1 − r)y) ≤ rf(x) + (1 − r)f(y) + Pα(r, ‖x − y‖) for x, y ∈ V and r ∈ [0, 1], where Pα: [0, 1] × [0, ∞) → [0, ∞) is a specific function dependent on α. We obtain three different estimations of Pα. This enables us to generalize some results concerning paraconvex and semiconcave functions.