The Schwarzian Derivative and Univalent Functions

Abstract

Abstract.In this paper we prove under certain conditions the function w=f(z) is univalent in |z|<l.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 |z|<l,/(0)=0,/'(0)==1; (b) f(z) has a pole at z=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 zplane. The simplest quantity of this type is the Schwarzian differential parameter {w, z} = (w"lw'Y -lOv'Vw')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 |a,|^l valid for functions w=f(z)=z~1+a0+ a,z+ • • • schlicht in the unit circle. Indeed, applying this inequality to the coefficient of z in the expansion of the schlicht function g(2) = _ nm -m2±_

Some Notes on Extensions of Basic Univalence Criteria

“…(3) If we consider m = 1 and α = 1 in Corollary 3.4, the inequalities (3.21) and (3.22) becomes (1.2) and then we obtain the univalence criterion due to Ozaki-Nunokawa [7]. …”

“…Three of the most important and known univalence criteria for analytic functions defined in the open unit disk were obtained by Nehari [4], OzakiNunokawa [7] and Becker [1]. Some extensions of these three criteria were given by (see [6,9,10,11,12,13] and [14]).…”

On a class of univalent functions

“…By virtue of a result due to Ozaki and Nunokawa [4], Obradovic et al [3] considered a class of univalent functions.…”

Criteria for univalence, starlikeness and convexity

“…According to a result due to Ozaki and Nunokawa [6], we have the inclusion U(λ) ⊂ S for 0 < λ ≤ 1, and from [3], we also have the inclusion P(2λ) ⊂ U(λ). In [4], the authors have shown that certain results obtained in [3] also hold if P(2λ) is replaced by U(λ).…”