Very Slowly Varying Functions

A real-valued function f of a real variable is said to be (p-slowly varying ((p-s .v.) if limn_ . rp (x) [ f (x + a) f (x)] = 0 for each a. It is said to be uniformly 9-slowly varying (u . (P-s .v .) if limn-. . sup, e r rp(x) ; f (x-a) f (x)I =0 for every bounded interval I. It is supposed throughout that rp is positive and increasing . It is proved that if w increases rapidly enough, then every rp-s .v . function fmust be u.9-s .v . and must tend to a limit at x . Regardless of the rate of increase of rp, a measurable function fmust be u.9-s .v . if it is w-s.v . Examples of pairs (,, f) are given that illustrate the necessity for the requirements on w and f in these results . Introduction The theory of slowly varying functions plays a role in analysis and number theory and has recently come to the fore in probability theory [3] . We consider here some simple, but basic questions about slowly varying functions . We prove four theorems and a lemma . I. Statement of Results Let cp be a positive non-decreasing real-valued function defined on [0, cc) and let f be any real-valued (not necessarily measurable) function defined on [0, oc) . The object of this paper is to study the condition for every a, cp (x) [f (x + u) f (x)] -0 as x , oo . (1.1) Whenever (1 .1) holds, we will say that f is cp-slowly varying, and abbreviate this by cp-s .v . If (1 .1) holds uniformly for a in each bounded interval, then we say that f is uniformly cp-slowly varying (u.(p-s .v .) . In other words, f is u.(p-s .v. if lim sup 9 (x) j f (x + (x) f (x)I = 0 for each bounded interval I . X-*c aer Throughout this paper, the words `measurable' and `measure' refer to Lebesgue measure . 1) The research of the first author was partially supported by NSF Grant GP 14986 . 2) The research of the third author was partially supported by a grant from the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant # AF OSR 68 1499 . Received April 6, 1971 2 J. Marshall Ash, P . Erdős and L . A . Rubel AEQ. MATH. Of course, iffis u .cp-s .v . then it is (p -s .v . The converse is 'almost' true . THEOREM 1 . If f is 9-slowly varying and measurable, then f is uniformly (Pslowly varying. THEOREM 2 . Iff is (p-slowly varying and if cp satisfies then f tends to a finite limit at co . Conversely, if or, equivalently, =o co