On Univalent k-Fold Symmetric Logharmonic Mappings

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