Close-to-starlike logharmonic mappings
A logharmonic mapping is defined as a solution of the nonlinear elliptic partial differential equation \[ \overline{f}_{\overline{z}}= a\frac{\overline{f}} {f} f_z, \] where \(a\) belongs to the linear space \(H(U)\) of all analytic functions defined on the unit disc \(U\) with \(| a(z)|< 1\), for all \(z\in U\). In this paper, close-to-starlike logharmonic mappings are defined and studied. Such mappings are of the form \(f= z| z|^{i\beta} h\overline{g}\) defined on \(U\) and can be written as the product of a logharmonic mapping with positive real part and a univalent starlike logharmonic mapping. The radius of univalence and the radius of starlikeness of these mappings are determined and the radii are shown to be exact. Also a representation theorem and sharp distortion theorems are proved.
scientific article; zbMATH DE number 894090