Improved Bohr radius for $ k$-fold symmetric univalent logharmonic mappings

TL;DR

This paper investigates $k$-fold symmetric univalent logharmonic mappings, deriving distortion bounds, area bounds, and improved Bohr radii, and introduces derivatives related to these mappings.

Contribution

It provides new distortion and area bounds, calculates improved Bohr radii for $k$-fold symmetric logharmonic mappings, and introduces pre-Schwarzian and Schwarzian derivatives for these functions.

Findings

Derived distortion bounds for the mappings.

Calculated improved Bohr radii.

Introduced pre-Schwarzian and Schwarzian derivatives for logharmonic mappings.

Abstract

We study the $k$-fold symmetric starlike univalent logharmonic mappings of the form $f(z)=zh(z)\overline{g(z)}$ in the unit disk $\mathbb{D}:= \lbrace z \in \mathbb{C}: |z|<1 \rbrace$ with several examples, where $h(z)=\exp \left(\sum_{n=1}^{\infty}a_{nk}z^{nk}\right)$ and $g(z)=\exp \left(\sum_{n=1}^{\infty}b_{nk}z^{nk}\right)$ are analytic in $\mathbb{D}.$ The distortion bounds of these functions are obtained, which give area bounds. Improved Bohr radii for this family are calculated. We also introduce the pre-Schwarzian and Schwarzian derivatives of logharmonic mappings that vanish at the origin.