On geometrical properties of logharmonic mappings

TL;DR

This paper investigates the geometric properties of starlike logharmonic mappings, determining radii of starlikeness and univalence, and exploring relations between different classes of these mappings.

Contribution

It introduces new radius estimates for starlike and close to starlike logharmonic mappings, enhancing understanding of their geometric behavior.

Findings

Radius of starlikeness for logharmonic mappings established

Relation between starlike logharmonic mappings of different orders derived

Radius of univalence for close to starlike mappings determined

Abstract

In this paper, we find the radius of the disk $\Omega _{r}$ such that every starlike logharmonic mapping $f(z)$ of order $\alpha ,$ is starlike in $% |z|\leq r$ with respect to any point of $\Omega _{r}.$ We also establish a relation between the set of starlike logharmonic mappings \ and the set of starlike logharmonic mappings of order alpha. Moreover, the radius of starlikeness and univalence for the set of close to starlike logharmonic mappings of order $\alpha $ is determined.