On concave univalent functions (Q5891550)

Let \(\alpha\in(1,2]\) be given. A univalent function \(f(z)= z+\cdots\) on the unit disk \(\mathbb{D}\) is said to belong to the class \(\text{Co}(\alpha)\) of concave functions if and only if \(f(1)=\infty\), the set \(\mathbb{C}\setminus f(\mathbb{D})\) is convex and unbounded, and the measure of the opening angle of \(f(\mathbb{D})\) at infinity is at most \(\pi\alpha\). The authors exploit relations between \(\text{Co}(\alpha)\) and classes of close-to-convex or Pareto functions to get bounds for various functionals over their class. Then they consider \(\text{Co}(\alpha)\) as a compact subset of the Hornich space [\textit{H. Hornich}, Monatsh. Math. 73, 36--45 (1969; Zbl 0167.42702)] to study extreme points.