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Letbe 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 functionin the class,As an important application, in the same paper, Koepf showed thatfor 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.

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