On the class gamma and related classes of functions

Abstract:

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 ε Γ0(g). Taking h = log f, we find that h 2 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.

Año de publicación:

2013

Keywords:

• The class gamma
• Differential equations
• Remainder terms
• Growth of functions
• Local uniform convergence
• Beurling variation

Fuente:

scopus