This paper studies the class consisting of univalent logharmonic mappings in the unit disk , where and are analytic in and is a normalized starlike analytic function. A representation theorem for these mappings is obtained, which yields sharp distortion estimates, and a sharp Bohr radius.

The classical Bohr inequality is studied in the Poincaré disk model of the hyperbolic plane. A sharp hyperbolic Bohr radius of is obtained for analytic self-maps of the hyperbolic unit disk. The hyperbolic Bohr radius is also investigated for the h-convex hull of a given set lying in an open half-plane of the unit disk. In this case, the sharp Bohr radius is also obtained. Additionally, the main theorem is applied to obtain a Bohr-type theorem for other hyperbolic regions.