Multidimensional analogs of the Fekete--Szeg\"{o} functional

In this paper we introduce the Fekete--Szeg\"{o} 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--Szeg\"{o} functional are either special cases or `components' of the mapping we introduce. The study involves the examination of transforms of the Fekete--Szeg\"o mapping under specific transformations applied to given holomorphic mappings. We show that for a mapping $f$, the third order Fr\'eshet derivative of the inverse mapping $f^{-1}$ and of elements of the semigroup generated by $f$ can be expressed in terms of the Fekete--Szeg\"{o} mapping. Estimates of the Fekete--Szeg\"o mapping over some subclasses of semigroup generators and of starlike mappings are also presented.