Properties of analytic functions with small schwarzian derivative

Abstract

Let be an analytic function defined in the unit disc. Gabriel has proved that if the Schwarzian derivative S( f,zf) of f is bounded by a constant, then fis a starlike function. We show that under the assumption that if S( f,z) and a 2, the second coefficient of f, are small then f is a strongly starlike function of order α. Some conditions found are best possible in certain sense. Moreover if S (f,z) is bounded by a smaller constant and together a 2 is also small, then f is a convex function.