The univalence of functions asymptotic to nonconstant logarithmic monomials

Introduction. The title refers to analytic functions s(x) which behave like nonconstant logarithmic monomials M(x) = cxmo(log x)ml ... (log, x)mr (where c is a complex number 0, the mi are real, and logN is the N-fold iterate of the principal determination of log) in the sense that s(x)/M(x)-*1 as x-* oo in the complex plane. DEFINITION. An analytic function E is said to -3O rapidly enough for M if E- O and (M/M')E'- O as x- oo. Theorem 2 states that if s = M(1 +E) where E-*O rapidly enough for M, then s is 1-1 in some neighborhood of infinity. The neighborhood bases for oo with which we shall be concerned are families F(a, ,B) whose elements are sector-like regions V(ax, ,B, t) defined as follows: Let -r?a< <3 r. Let t(3) be a real-valued function defined and bounded below on some subinterval (0, y) of (0, (j3-a)/2). Let T(ax+S, B-3, t(S)ei) be the sector {z:ax+3