O-regularly Varying Functions and Some Asymptotic Relations

Dragan Đurčić

We prove that in the class of measurable positive functions defined on the interval $I_a = [\,a,+\infty )$ $(a > 0)$, the class of functions which preserve the strong asymptotic equivalence on the set of functions $\{x \,\colon I_a \mapsto \Bbb R^+ ,\, x(t) \to +\infty, t \to +\infty \}$, is a class of $\Cal O$--regularly varying functions with continuous index function. We also prove a representation theorem for functions from this class, and a morphism-theorem for some asymptotic relations.