An extension of a theorem of Perron

We give sufficient conditions for a linear differential equation of order n to have a solution whose logarithmic derivative is asymptotically in a small neighbourhood of a given constant. This asymptotic behavior is specified by means of a general comparison function $\varphi $. An appropriate formulation of the growth properties of $\varphi $ is introduced. The differential equation is regarded as a perturbation of some constant coefficient equation. Our smallness conditions permit conditional convergence of some corresponding improper integrals.