A class of starlike functions
Let \(S^*\) be the usual class of starlike functions, i.e. \(f(z)\) of the form \(z+a_ 2 z^ 2\dots\) which are analytic and univalent in the unit disc \(U\) and which map \(U\) onto a domain starshaped with respect to the origin. Let \(0<p<1\) and \(S^*(p)\) be the class of functions analytic in \(U\) such that \(f(0)=1\), \(f(p)=0\), and \(f\) maps \(U\) univalently onto a domain starshaped with respect to the origin. In this paper, a representation formula is found for the closed convex hull of \(S^*(p)\). The extreme points are the functions \(F_ x(z)=-(z-p)(1-pz)/p(1-xz)^ 2\) where \(| x|=1\). An integral mean result is proved: If \(f(z)\in S^*(p)\) and \(\lambda\) real, \(\int^ \pi_{-\pi} | f(re^{i\theta})|^ \lambda d\theta\leq \int^ \pi_{-\pi} | F_{-1}(re^{i\theta})|^ \lambda d\theta\). The proof makes use of a result which is quite interesting in its own right: \(f\in S^*(p)\) if and only if there exists a \(g\in S^*\) such that \(f(z)=-(z-p)(1- pz)g(z)/pz\).
scientific article; zbMATH DE number 218646