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 $${\mathbb {C}}^{n}.$$ C n . More precisely, the article concerns a Bavrin’s family of functions defined on a bounded complete n-circular domain $${\mathcal {G}}$$ G of $${\mathbb {C}}^{n}$$ C n and a family of biholomorphic mappings on the Euclidean open unit ball in $${\mathbb {C}}^{n}.$$ C n . The presented results include some estimates of a combination of the Fréchet differentials at the point $$z=0,$$ 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 $${\mathbb {C}}^{n},$$ 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.

On the coefficient inequalities for some classes of holomorphic mappings in complex Banach spaces

“…Using the above equalities, (3)(4)(5)(6)(7)(8) and (3)(4)(5)(6)(7)(8)(9), we obtain all of the desired conclusions about Theorems 3.1. The example which shows the sharpness of Theorem 3.1 is the same as the mapping defined in [37].…”

“…By using (3-10), (3)(4)(5)(6)(7)(8)(9)(10)(11) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12), we obtain k ∈ S * . Let s(x) = (D F(x)) −1 H (x), and let…”

Note on the Fekete–Szegö Problem for Spirallike Mappings in Banach Spaces

“…In the multidimensional settings, several analogs of the Fekete-Szegö functional have been suggested starting since 2014 (see, for instance, [4,9,15,16], [29]- [31]). To present them more specifically, we need the following notations.…”

“…Several modifications to the classical Fekete-Szegö functional have been proposed. Their estimates over various classes of holomorphic mappings were studied (see, for instance, [4,8,9,15,16,18] and [29]- [32]).…”

On refinements of the Fekete-Szegö inequality for a class of holomorphic mappings associated with spirallike mappings

“…After that, the Fekete and Szegö inequalities are established for well-known subclasses of biholomorphic mappings in higher dimensions. For more details, we refer the reader to the papers ([1]- [2], [7]- [8], [12], [15]- [16]).…”