Generalized regular variation of second order

Assume that for a measurable function f on (0, infinity) there exist a positive auxiliary function a(t) and some gamma is an element of R such that phi(x) = lim(t-->infinity)(f(tx) - f(t))/a(t) = integral x-1 s(gamma-1)ds, x > 0. Then f is said to be of generalized regular variation. In order to control the asymptotic behaviour of certain estimators for distributions in extreme value theory we are led to study regular variation of second order, that is, we assume that lim(t-->infinity)(f(tx) - f(t) - a(t)phi(x))/a(1)(t) exists non-trivially with a second auxiliary function a(1)(t). We study the possible limit functions in this limit relation (defining generalized regular variation of second order) and their domains of attraction. Furthermore we give the corresponding relation for the inverse function of a monotone f with the stated property. Finally, we present an Abel-Tauber theorem relating these functions and their Laplace transforms.