Support points of some classes of analytic and univalent functions

TL;DR

This paper investigates the properties and support points of specific classes of analytic and univalent functions within the unit disk, focusing on classes defined by geometric and differential inequalities.

Contribution

It characterizes support points for classes nd nd explores their geometric and analytic properties, providing new insights into their structure.

Findings

Support points identified for classes nd nd their extremal functions characterized.

Derived coefficient bounds and growth estimates for functions in these classes.

Established inclusion relations and boundary behavior of the classes.

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 $${\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2} \quad\mbox{ for } z\in\mathbb{D}.$$