A Characterization of concave mappings

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 $1+z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)$, 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.