On extreme points and support points of the family of starlike functions of order \(\alpha\) (Q1084543)

Let \(S^*(\alpha)\) denote the set of starlike functions of order \(\alpha\), that is, \(f\in S^*(\alpha)\) provided that f is holomorphic in the open unit disk, \[ f(0)=0,\quad f'(0)=1\quad and\quad Re\{(zf'(z))/(f(z))\}>\alpha \] for \(| z| <1\). Also let \(s[S^*(\alpha)]\) denote the set of functions which are subordinate to some member of \(S^*(\alpha).\) The authors find the closed convex hull of \(s[S^*(\alpha)]\) where \(0<\alpha <\) and the extreme points are determined to be \[ f(z)=(xz)/((1-yz)^{2(1-\alpha)})\text{ where } | x| =| y| =1. \] These results were proved earlier when \(\alpha\leq 0\) and \(\alpha =\) by \textit{D. J. Hallenbeck} and the reviewer [Pac. J. Math. 50, 455-468 (1974; Zbl 0258.30015); Proc. Am. Math. Soc. 46, 59-64 (1974; Zbl 0298.30011)]. For \(\alpha <\), the authors also prove that the functions described above give the set of support points of \(s[S^*(\alpha)]\) for ''nontrivial'' functionals. The set of supports points is also found for the related class of functions having the representation \[ f(z)=\int_{T}(xz)(1-yz)^{-p})d\mu (x,y)\text{ where } p>0 \] and \(\mu\) is a probability measure on the torus T.