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 $\mathbb{D}:= \lbrace z \in \mathbb{C}: |z|<1 \rbrace$ 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.