A generalization of slowly varying functions

Abstract:

This note establishes that if the main part of the definition of a slowly varying function is relaxed to the requirement that lim ${\sup _{x \to \infty }}\psi (\lambda x)/\psi (x) < \beta < \infty$ for each $\lambda > 0$, then $\psi (x) = L(x)\theta (x)$, where $L$ is slowly varying and $\theta$ is bounded. This is done by obtaining a representation for the function $\psi$.