Order of starlikeness for multipliers of univalent functions (Q762641)

A function \(f(z)=z-\sum^{\infty}_{n=2}a_ nz^ n\), \(| z| <1\) and \(a_ n\geq 0\) is in \(F(\{b_ n\})\) if \(\sum^{\infty}_{n=2}b_ na_ n\leq 1\) for the sequence of positive numbers \(\{b_ n\}\). The author shows that \(F(\{b_ n\})\) is a convex family of univalent functions if and only if \(b_ n\geq n\) for each n and that \((z-z^ n/b^ n)\) and the identity are the extreme points of the class; distortion and other mapping properties of the class are given. He also defines and examines the classes corresponding to \(F(\{b_ n\})\) when the coefficients \(a_ n\) are arbitrary complex numbers.