The Schwarzian derivative and schlicht functions

It is customary to formulate the inequalities of the "Verzerrungssatz" type for analytic functions w-f (z), schlicht in the unit circle, with reference to a specific normalization. The two normalizations mainly used are: (a) f (z) is finite in \ <1, / (O) =0, /' (O) = 1; (b) (z) has a pole at 2 = 0 with the residue 1. If we want to obtain inequalities which are independent of any particular normalization, we have to use quantities which are invariant with regard to an arbitrary linear transformation of the z-plane. The simplest quantity of this type is the Schwarzian differential parameter / w"\' 1 / w"\ 2 also called the Schwarzian derivative of w with regard to z.