Some properties of univalent log-harmonic mappings

TL;DR

This paper investigates properties of log-harmonic starlike mappings, including representation, distortion, coefficient estimates, Bohr's radius, and introduces derivatives and Bloch's norm, expanding understanding of their geometric behavior.

Contribution

It provides new theorems and estimates for log-harmonic mappings, generalizing previous results and introducing new concepts like pre-Schwarzian and Schwarzian derivatives for these functions.

Findings

Representation and distortion theorems established

Coefficient estimates and Bohr's radius determined

Inner mapping radius and derivatives for log-harmonic mappings analyzed

Abstract

We determine the representation theorem, distortion theorem, coefficients estimate and Bohr's radius for log-harmonic starlike mappings of order $\alpha$, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of $1$-slit log-harmonic mappings. Finally, we introduce pre-Schwarzian, Schwarzian derivatives and Bloch's norm for non-vanishing log-harmonic mappings, several properties related to these are also obtained.