Convex hulls and extreme points of classes of multivalent functions defined by gap power series (Q761562)

The authors determine the closed convex hulls and extreme points of various classes of p-valent functions whose power series have k-fold symmetric gaps. For example if \(f(z)=z^ p+\sum^{\infty}_{m=1}a_{mk+p}z^{mk+p}\) and \(Re zf'(z)/f(z)>p\alpha\) where \(\alpha <1\) then f is p-valently starlike of order \(\alpha\). The authors prove that the extreme points of the closed convex hull of this class are given by the set of functions \(\{(z^ p/(1-xz^ k)^{2p(1-\alpha)/k}):\quad | x| =1\}.\) They also give some applications to coefficient estimates. The techniques used throughout are standard.