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- Title
Generalized approximate midconvexity.
- Authors
Tabor, Jacek; Tabor, Józef
- 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 (These character(s) cannot be represented in ASCII text) . 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 Ȇ V and r Ȇ [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.
- Subjects
GENERALIZED inverses of linear operators; APPROXIMATION theory; GENERALIZED estimating equations; THEORY of distributions (Functional analysis); GENERALIZED spaces; BERNSTEIN polynomials; CONVEX sets
- Publication
Control & Cybernetics, 2009, Vol 38, Issue 3, p655
- ISSN
0324-8569
- Publication type
Article