<p>In this paper we introduce the Fekete–Szegö type mapping in the open unit ball of a complex Banach space. All previously studied modifications of the Fekete–Szegö functional are either special cases or ‘components’ of the mapping we introduce. For the first time, we analyze the geometric properties of the Fekete–Szegö mapping under classical transformations previously studied only in the one-dimensional case. Estimates of the Fekete–Szegö mapping over some subclasses of semigroup generators and of starlike mappings are also presented.</p>