Fekete–Szegö problem for Bavrin’s functions and close-to-starlike mappings in $${\mathbb {C}}^{n}$$
Renata Długosz, Piotr Liczberski
Abstract
Abstract The paper is devoted to the study of a family of complex-valued holomorphic functions and a family of holomorphic mappings in $${\mathbb {C}}^{n}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> More precisely, the article concerns a Bavrin’s family of functions defined on a bounded complete n -circular domain $${\mathcal {G}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> of $${\mathbb {C}}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> and a family of biholomorphic mappings on the Euclidean open unit ball in $${\mathbb {C}}^{n}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The presented results include some estimates of a combination of the Fréchet differentials at the point $$z=0,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> of the first and second order for Bavrin’s functions, also of the second and third order for biholomorphic close-to-starlike mappings in $${\mathbb {C}}^{n},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> 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.