Abstract

We study some properties of functions that satisfy the condition f′(x) = o (\frac{f(x)}{x}), for x → ∞, i.e., lim_{x→∞} \frac{f′(x)}{\frac{f(x)}{x}} = 0. We call these

“functions of slow increase”, since they satisfy the condition lim_{x→∞} \frac{f(x)}{x^{α}} = 0 for all α > 0. A typical example of a function of slow increase is the function f(x) = log x. As an application, we obtain some general results on sequence A_{n} of positive integers that satisfy the asymptotic formula A_{n} ∼ n^{s}f(n), where f(x) is a function of slow increase.