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On the class gamma and related classes of functions

Omey Edward (HUB, Brussels, Belgium)

The gamma class Γα(g) consists of positive and measurable functions that satisfy f(x + yg(x))/f(x) → exp(αy). In most cases the auxiliary function g is Beurling varying and self-neglecting, i.e., g(x)/x → 0 and gΓ0(g). Taking h = log f, we find that hEΓα(g, 1), where EΓα(g, a) is the class of positive and measurable functions that satisfy (f(x + yg(x))− f(x))/a(x) → αy. In this paper we discuss local uniform convergence for functions in the classes Γα(g) and EΓα(g, a). From this, we obtain several representation theorems. We also prove some higher order relations for functions in the class Γα(g) and related classes. Two applications are given.

Keywords: Beurling variation, the class gamma, local uniform convergence, remainder terms, differential equations, growth of functions