Regularity Conditions for Approximately Convex Functions on Abelian Topological Groups

Abstract

Let ${(G, +)}$ be an Abelian topological group, D be a subset of G and let a function ${\alpha: D - D \rightarrow \mathbb{R}}$ be locally bounded above at zero. A function ${f: D \rightarrow \mathbb{R}}$ we call ${\alpha}$-convex if

$$f(z) \leq \frac{f(x)+f(y)}{2}+\alpha(x-y)$$

for all ${x,y,z \in D}$ such that ${x+y=2z}$. We prove that if ${\alpha(0)=0}$, ${\alpha}$ is continuous at zero, D is open and connected, f is ${\alpha}$-convex and locally bounded above at a point then f is locally uniformly continuous. We show that the same is true if we replace the assumption that f is locally bounded above at a point by assumption that f is Haar measurable or Baire measurable. We give also Ostrowski-type and Mehdi-type theorem for such functions.