Support points of some classes of analytic and univalent functions

Abstract

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ satisfying $f(0)=0$ and $f'(0)=1$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying $$\left|f'(z)\left(\frac{z}{f(z)}\right)^2-1 \right|< 1 \quad\mbox{ for } z\in\mathbb{D},$$ and $\mathscr{G}$ denote the class of functions $f\in \mathcal{A}$ satisfying $${\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2} \quad\mbox{ for } z\in\mathbb{D}.$$