Multidimensional analogs of the Fekete--Szegö functional

In this paper we introduce the Fekete–Szegö type mapping in the open unit ball of a complex Banach space. % and study its geometric and analytical properties. All previously studied modifications of the Fekete–Szegö functional are either special cases or `components’ of the mapping we introduce. The study involves the examination of transforms of the Fekete–Szegö mapping under specific transformations applied to given holomorphic mappings. We show that for a mapping $f$, the third order Fréshet derivative of the inverse mapping $f^{-1}$ and of elements of the semigroup generated by $f$ can be expressed in terms of the Fekete–Szegö mapping. Estimates of the Fekete–Szegö mapping over some subclasses of semigroup generators and of starlike mappings are also presented.