Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 273.26001
Autor: Ash, J.Marshall; Erdös, Paul; Rubel, L.A.
Title: Very slowly varying functions. (In English)
Source: Aequationes Math. 10, 1-9 (1974).
Review: Let \phi be a positive non-decreasing real valued function defined on [0, oo), and let f be any real valued function defined on [0, oo). We say that f is \phi-slowly varying if \phi (x)[f(x+\alpha)-f(x)] ––> 0 as x ––> oo for each \alpha. We say that f is uniformly \phi-slowly varying if \sup {\phi (x) |f(x+\alpha)-f(x)|: \alpha in I } ––> 0 as x ––> oo for every bounded interval I. We state here five theorems that will be proved later in a longer communication. We also pose one question that seems to be difficult. Theorem 1. If f is \phi-slowly varying and if sum i/ \phi (n) < oo, then f tends to a finite or infinite limit at oo. Theorem 2. If f is \phi-slowly varying and measurable, then f is uniformly \phi-slowly vayring. Theorem 3. Let f be \phi-slowly varying and let \beta (x) = sum ooj = 0 1/ \phi (x+j). If \phi (x) \beta (x) is bounded, then f must be uniformly \phi-slowly varying. Theorem 4. Suppose that sum 1/ \phi (n) < oo and that \phi (x+1)/ \phi (x) ––> 1 as x ––> oo. Then there exists a function f that is \phi-slowly varying but not uniformly \phi-slowly varying. Theorem 5. Let \beta (x) be the function of Theorem 4, and suppose that \phi (x) \beta (x) is unbounded, but that \phi (x) \beta (x) = o(x) as x ––> oo. Then there exists a function f that is \phi-slowly varying but not uniformly \phi-slowly varying. Question. Does there exist a function f such that x[f(x+\alpha)-f(x)] ––> 0 as x ––> oo for each \alpha but \sup {|f(x+\alpha)-f(x)|: \alpha in [0,1] } (not)––> 0 as x ––> oo?
Classif.: * 26A12 Rate of growth of functions of one real variable
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