  In this paper we study sharp estimates of pre-Schwarzian derivatives of
functions belonging to the Nehari-type classes by using techniques from
differential equations. In the sequel, we also see that a solution of a complex
differential equation has a special form in terms of ratio of hypergeometric
functions resulting to an integral representation. Finally, we attempt to study
those univalent functions in the unit disk for which the image domain is an
unbounded John domain.
