On extreme points of subordination families with a convex majorant
This paper contains a number of contributions to the study of the set of extreme points Es(F) of the family of functions s(F) which are subordinate to F, where F is a convex analytic mapping of \(\Delta =\{z:| z| <1\}\) onto D. The main results are quoted below where throughout it is assumed that \(f\in s(F).\) 1. If the line segment PQ\(\subset \partial D\) and \(\partial f(\Delta)\cap \partial D\subset PQ\setminus \{P,Q\}\) then \(f\not\in Es(F).\) 2. If \(f\in Es(F)\) then \(E\partial D\cap \overline{\{f(e^{i\theta})\}}\neq \emptyset\). Here \(E\partial D\) denotes the set of extreme points of \(\partial D\), \(f(e^{i\theta})\equiv \lim_{r\to 1-}f(re^{i\theta})\) and \(\{f(e^{i\theta})\}\) denotes the set of all radial limits of f. 3. Suppose that f is univalent, f(\(\Delta)\) is a Jordan domain and \(f\in Es(F)\). Then there is a point \(P\in E\partial D\) and an arc \(\Gamma\) \(\subset \partial \Delta\) such that \(P\in \overline{\{f(e^{i\theta}):\theta \in \Gamma \}}\) and for each subarc \(\Gamma '\) of \(\Gamma\) for which \(P\in \overline{\{f(e^{i\theta}):\theta \in \Gamma '\}}\) we have \[ \int_{\Gamma '}\log \lambda (\theta)d\theta =-\infty. \] Here \(\lambda\) (\(\theta)\) denotes the distance bewteen \(f(e^{i\theta})\) and \(\partial D.\) 4. If \(\partial D\) is suitably smooth, then \(Es(F)=\{f:f=F\circ \phi\), \(\phi \in EB_ 0\}\). Here \(EB_ 0\) denotes the set of extreme points of \(B_ 0\), the set of functions \(\omega\) analytic in \(\Delta\) and satisfying \(| \omega (z)| <1\) for \(| z| <1\) and \(\omega (0)=0.\) A general reference for earlier results in this area is [``Linear problems and convexity techniques in geometric function theory'', \textit{D. J. Hallenbeck} and \textit{T. H. MacGregor} (1984; Zbl 0581.30001)].
scientific article; zbMATH DE number 4136298