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

We study the $k$-fold symmetric starlike univalent logharmonic mappings of the form $f(z)=zh(z)\overline{g(z)}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ with several examples, where $h(z)=\exp \left({\sum }_{n=1}^{\mathrm{\infty }}{a}_{nk}{z}^{nk}\right)$ and $g(z)=\exp \left({\sum }_{n=1}^{\mathrm{\infty }}{b}_{nk}{z}^{nk}\right)$ are analytic in $\mathbb{D}\mathrm{.}$ 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.