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.
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The research of the first author was partially supported by NSF Grant # GP 14986.
The research of the third author was partially supported by a grant from the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant # AF OSR 68 1499.
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Ash, J.M., Erdös, P. & Rubel, L.A. Very slowly varying functions. Aeq. Math. 10, 1–9 (1974). https://doi.org/10.1007/BF01834775
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DOI: https://doi.org/10.1007/BF01834775