Some analytical properties of γ-convex functions on the real line

Abstract

This paper deals with the analytical properties of γ-convex functions, which are defined as those functions satisfying the inequalityf(x ^′₁ )+f(x ^′₂ )≤f(x ₁)+f(x ₂), forx ^′ _i ∈[x ₁,x ₂], |x _i −x ^′ _i |=γ, i=1,2, whenever |x ₁−x ₂|>γ, for some given positive γ. This class contains all convex functions and all periodic functions with period γ. In general, γ-convex functions do not have ideal properties as convex functions. For instance, there exist γ-convex functions which are totally discontinuous or not locally bounded. But γ-convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length γ. It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of γ-convex functions on the real line. However, γ-convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of γ-convexity are given. Ways for generating and representing γ-convex functions are described.