Von Mises-type conditions in second order regular variation (Q1916723)

A function \(f\) is of extended regular variation of second-order if \[ \lim_{t\to\infty} {f(tx)-f(t)-a(t)(x^\gamma-1)/\gamma\over c(t)}= \int^x_1 s^{\gamma-1}\int^s_1 u^{\rho-1}du ds \] for some (positive) functions \(a\) and \(c\). Here \(\gamma\in\mathbb{R}\) and \(\rho\leq0\). The author gives sufficient conditions involving derivatives.