Characterizing Rotationally Typically Real Logharmonic Mappings

TL;DR

This paper investigates a specific class of logharmonic mappings with typically real associated functions, providing their structural decomposition, geometric properties, and bounds on starlikeness and arclength.

Contribution

It introduces a novel factorization of these mappings into integral-represented components and analyzes their geometric and univalence properties.

Findings

Decomposition into two integral-represented logharmonic mappings

Determined the radius of starlikeness

Provided an upper bound for arclength

Abstract

This paper treats the class of normalized logharmonic mappings f(z) = zh(z)bar{g(z)} in the unit disk satisfying {\phi}(z) = zh(z)g(z) is analytically typically real. Every such mapping f is shown to be a product of two particular logharmonic mappings, each of which admits an integral representation. Also obtained is the radius of starlikeness and an upper estimate for arclength. Additionally, it is shown that f maps the unit disk into a domain symmetric with respect to the real axis when it is univalent and its second dilatation has real coefficients.