A Characterization of concave mappings

TL;DR

This paper characterizes concave univalent functions mapping the unit disk onto domains with convex complements, analyzing their properties through the real part of a specific expression and the Schwarzian derivative.

Contribution

It provides a new characterization of concave mappings using the real part of $1 + zf''(z)/f'(z)$ and explores the influence of the Schwarzian derivative for bounded convex domains.

Findings

Characterization of concave mappings via the real part of $1 + zf''(z)/f'(z)$

Role of Schwarzian derivative in bounded convex domains

Insights into the behavior of univalent functions with convex complements

Abstract

This study focuses on Concave mappings, a class of univalent functions that exhibit a unique property: they map the unit disk onto a domain whose complement is convex. The main objective of this work is to characterize these mappings in terms of the real part of the expression $1 +zf''(z)/f'(z)$, considering scenarios where the omitted convex domain is either bounded or unbounded. In the case of a bounded convex domain, we investigate the pivotal role played by the Schwarzian derivative and the order of the functions in understanding the behavior and properties of these mappings.