Fekete–Szegö problem for Bavrin’s functions and close-to-starlike mappings in {mathbb {C}}^{n}

Abstract

The paper is devoted to the study of a family of complex-valued holomorphic functions and a family of holomorphic mappings in {mathbb {C}}^{n}. More precisely, the article concerns a Bavrin’s family of functions defined on a bounded complete n-circular domain {mathcal {G}} of {mathbb {C}}^{n} and a family of biholomorphic mappings on the Euclidean open unit ball in {mathbb {C}}^{n}. The presented results include some estimates of a combination of the Fréchet differentials at the point z=0, of the first and second order for Bavrin’s functions, also of the second and third order for biholomorphic close-to-starlike mappings in {mathbb {C}}^{n}, respectively. These bounds give a generalization of the Fekete–Szegö coefficients problem for holomorphic functions of a complex variable on the case of holomorphic functions and mappings of several variables.