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\begin{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.
\end{abstract}

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