Radii of Concavity for Subclasses of Univalent Functions Associated with the Exponential Mapping

Abstract

We determine the radii of concavity for the classes ${\mathcal{S}}_e^{\ast }$ and ${\mathcal{C}}_e$ of univalent functions associated with the exponential mapping $e^z$. Under the geometric framework of Avkhadiev--Wirths for conformal mappings with unbounded convex complements of opening angle $\pi A$ ($A\in (1,2]$), the radii are characterized by explicit transcendental equations. Sharpness is established globally by constructing explicit univalent extremal functions related to the disk automorphism ${\omega }_0(z)=-z$, and the strict monotonicity of these radii with respect to the parameter $A$ is verified numerically.