On the class gamma and related classes of functions

The gamma class (g) consists of positive and measurable func- tions that satisfy f(x + yg(x))/f(x) ! exp(y). In most cases the auxil- iary function g is Beurling varying and self-neglecting, i.e., g(x)/x ! 0 and g 2 0(g). Taking h = logf, we find that h 2 E (g,1), where E (g,a) is the class of positive and measurable functions that satisfy (f(x + yg(x)) f(x))/a(x) ! y. In this paper we discuss local uniform convergence for functions in the classes (g) and E (g,a). From this, we obtain several representation theorems. We also prove some higher order relations for func- tions in the class (g) and related classes. Two applications are given.