Abstract
In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which elements satisfy also a (j, k)-symmetry condition. As application of these results there is given the solution of a Fekete–Szegö type problem for a family of normalized biholomorphic starlike mappings in {mathbb {C}}^{n}.
Highlights
By C, R, Z, N0, N1, N2 let us denote the sets of complex numbers, real numbers, all integers, nonnegative integers, positive integers and the integers not smaller than 2, respectively
We say that a domain G ⊂ Cn, n ∈ N1, is complete n-circular if z = (z1 1, ..., zn n) ∈ G for each z = (z1, ..., zn) ∈ G and every = ( 1, ..., n) ∈ Un, where U is the unit disc {ζ ∈ C ∶ |ζ | < 1}
That the sharpness in the Bavrin’s result was understood as follows: There exists a bounded complete 2-circular domain G ⊂ C2 and a function f ∈ BG which realizes the equality in the above inequality
Summary
By C, R, Z, N0, N1, N2 let us denote the sets of complex numbers, real numbers, all integers, nonnegative integers, positive integers and the integers not smaller than 2, respectively. That the sharpness in the Bavrin’s result was understood as follows: There exists a bounded complete 2-circular domain G ⊂ C2 and a function f ∈ BG which realizes the equality in the above inequality. Note that the above unique decomposition (1.2) of functions was used in [20] to solve some functional equations, in [21] to construction a semi power series and in [22] to obtain a uniqueness theorem of Cartan type for holomorphic mappings in Cn. We close this section with the following Golusin’s [10] result, very useful in the proof of the first result of Fekete–Szegö type for holomorphic functions of several complex variables.
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