The Fekete and Szegö inequality for a class of holomorphic mappings on the unit polydisk in n?> and its application

Abstract

Let be the familiar class of normalized close-to-convex functions in the unit disk. In Koepf [On the Fekete-Szegö problem for close-to-convex functions. Proc Amer Math Soc. 1987;101:89–95], Koepf proved that for a function in the class , As an important application, in the same paper, Koepf showed that for close-to-convex functions. In this paper, we extend the above results to a subclass of close-to-quasi-convex mappings of type B defined on the unit polydisc in , and establish the sharp difference bound for the second and third coefficients of homogeneous expansions for this class of holomorphic mappings. The results presented here would provide a new path for solving the Bieberbach conjectures in several complex variables.