JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 34, 302-315 (1971) Slowly Varying Functions and Asymptotic Relations R. BOJANIC” Department of Mathematics, Ohio State University, Columbus, Ohio 43210 AND E. SENETA Australian National University Submitted by N. G. deBrut@ Received March 20, 1970 1. INTRODUCTION AND RESULTS We shall say that L is a slowly varying (SV) function if L is a real-valued, positive, and measurable function on [A, CO), A > 0, and if lim Lo = 1 *+a L(x) (1.1) for every X > 0. The most important properties of SV functions may be stated as follows: UNIFORM CONVERGENCE THEOREM. If L is a SV function, then for every [u, b], 0 < a < b < CO, the relation (1.1) holds uniform@ with respect to x E [a, b]. REPRESENTATION THEOREM. If L is a SV function, then there exists a positive number B > A such that for all x > B we have L(x) = exp (~(5) + /I $ dt) , * The author gratefully acknowledges support by the National Science Foundation under grant GP-9493. 302