Radius problems associated with pre-Schwarzian and Schwarzian derivatives

摘要

Some of important univalence criteria for a non-constant meromorphic function f(z) on the unit disk D involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss the problem of finding the largest possible r ∈ (0,1) for which the pre-Schwarzian norm of the dilation r~(-1)f(rz) is not greater than a prescribed number for normalized univalent functions f(z) in the unit disk. Similar results concerning the Schwarzian derivative are also obtained.