NEW REPRESENTATION OF SLOWLY VARYING FUNCTIONS

Abstract

For a slowly varying function $L(t)$ a new integral representation is obtained:

$$L(t) = \mu(t)\int\limits_t_0 ^t b(x)d\ln x, t\geq t_{_}0> 0, $$

where $\mu(t)$ is measurable on $[t_0, +\infty), b(t)$ is continuous on $[t_0, +\infty)$ and

$\lim \limits_{t\ringrow + \infty}(b(t) / L(t))= 0.$

This representation allows to generalize D.D. Adamovich’s classical result on equivalent slowly varying functions and to extend the statement of A. A. Goldberg theorem.