  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.
