Very slowly varying functions

Abstract

A real-valued functionf of a real variable is said to beϕ-slowly varying (ϕ-s.v.) if lim _x→∞ ϕ(x) [f(x+α)−f(x)]=0 for each α. It is said to be uniformlyϕ-slowly varying (u.ϕ-s.v.) if lim _x→∞ sup _{α ∈ I} ϕ(x) |f(x+α)−f(x)|=0 for every bounded intervalI.

It is supposed throughout that ϕ is positive and increasing. It is proved that ifϕ increases rapidly enough, then everyϕ-s.v. functionf must be u.ϕ-s.v. and must tend to a limit at ∞. Regardless of the rate of increase ofϕ, a measurable functionf must be u.ϕ-s.v. if it isϕ-s.v. Examples of pairs (ϕ,f) are given that illustrate the necessity for the requirements onϕ andf in these results.