The Schwarzian derivative and univalent functions

In this paper we prove under certain conditions the function w=f(z) is univalent in IzI <1. 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 Izl<1,f(O)=O,f'(O)=1; (b)f(z) has a pole at z=O 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 zplane. The simplest quantity of this type is the Schwarzian differential parameter {w, z} = (w'/w')'1(,V"IWI)2 also called the Schwarzian derivative of w with regard to z. It is easy to obtain an upper bound for {w, z} by a simple transformation of the classical inequality Ia, ?1_ valid for functions w=f(z) =z-1 +ao+ alz+ * * * schlicht in the unit circle. Indeed, applying this inequality to the coefficient of z in the expansion of the schlicht function g(z)f'(x)(= x2) f((Z + X)/(1 + XZ)) f(x) I + X-I f(X) (I-lXl2 -(1( ( ] + .X.. IxI <1, we obtain I{w, z}I<6/(I-Iz12)2 [3, p. 226]. It is known that by replacing the number 6 in this inequality by 2, this necessary condition for the univalence off(z) in Izl <1 becomes sufficient (141, 121, 151, 1l1). Received by the editors March 8, 1971. AMS 1969 subject classfiWcations. Primary 3040, 3042.