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 → ∞ ψ ( λ x ) / ψ ( x ) > β > ∞ {\sup _{x \to \infty }}\psi (\lambda x)/\psi (x) > \beta > \infty for each λ > 0 \lambda > 0 , then ψ ( x ) = L ( x ) θ ( x ) \psi (x) = L(x)\theta (x) , where L L is slowly varying and θ \theta is bounded. This is done by obtaining a representation for the function ψ \psi .