Growth properties of functions in Hardy fields

This paper continues the author's earlier work on the notion of rank in a Hardy field. Further results are given on functions in Hardy fields of finite rank, including extensions of Hardy's results on the rates of growth of his logarithmicoexponential functions. 1. Comparability classes. Nonzero elements a, ,B of an ordered abelian group are called comparable if there are positive integers m, n such that mlal > IIlt and n1,I1 > lal. Nonzero elements f, g of a Hardy field k are accordingly called comparable if both lim, -. f(x) and limxX g(x) are 0 or + no and their rates of approach to 0 or + xo or -xo as x -+ no are comparable, that is if the nonzero elements v(f ), v(g) of the value group v(k*) of k are comparable, as in [5, ?3]. (We assume some familiarity with [5], whose notation we employ.) More generally, germs f, g of continuous real-valued functions on positive half-lines in R which are nowhere zero on some half-line and are such that lim x X , f(x) and lim x X g(x) are either 0, or + no, or o, will be called comparable if, on some half-line, each of If l, lgl is bounded above and below by suitable integral powers of the other. Comparability is an equivalence relation among such germs. In particular, comparability is an equivalence relation among all nonzero elements f of Hardy fields with the property that lim5 f (x) is one of 0, + oo, or -oo, that is among all nonzero elements f such that v(f ) '$ 0. For f a germ of a nowhere zero continuous real-valued function on a positive half-line that approaches 0, + oo, or -oo as x -* + no, denote the comparability class of f by Cl(f). Noting that f, -f, 1/f, and -1/f are comparable and that precisely one of these is infinitely increasing (that is, approaches + no), we define Cl(f) Cl(v) if and only if v(u'/u) < v(v'/v), which holds if and only if v(log u) < v(log v). Received by the editors September 20, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 34E05, 41A60; Secondary 12H05, 13N05, 26A12. Research supported by National Science Foundation grant number DMS-8303286. ?31987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page