On approximately convex functions

A real valued function f defined on a real interval I is called (e, δ)-convex if it satisfies f(tx+(1-t)y) ≤ t f(x) + (1-t)f(y)+et(1-t)|x-y|+δ for x, y ∈ I, t ∈ [0, 1. The main results of the paper offer various characterizations for (e, δ)-convexity. One of the main results states that f is (e, δ)-convex for some positive e and δ if and only if f can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case e = 0, the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called δ-convexity.