Abstract:

We study some properties of functions that satisfy the condition $f'(x)=o\left(\frac{f(x)}{x}\right)$, for $x\rightarrow \infty$, i.e., $\lim_{x\rightarrow \infty}\frac{ f'(x)}{\frac{f(x)}{x}}= 0$. We call these ``functions of slow increase'', since they satisfy the condition $\lim_{x\rightarrow \infty}\frac{f(x)}{x^{\alpha}} =0$ for all $\alpha>0$. 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 $A_n \sim n^s f(n)$, where $f(x)$ is a function of slow increase.