Some results of Fekete–Szegö type for Bavrin’s families of holomorphic functions in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^{n}$$\end{document}

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 Cn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^{n}.$$\end{document}


Introduction
By ℂ, ℝ, ℤ, ℕ 0 , ℕ 1 , ℕ 2 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 ⊂ ℂ n , n ∈ ℕ 1 , is complete n-circular if z = (z 1 1 , ..., z n n ) ∈ G for each z = (z 1 , ..., z n ) ∈ G and every = ( 1 , ..., n ) ∈ U n , where U is the unit disc { ∈ ℂ ∶ | | < 1} . From now on by G will be denoted a bounded complete n-circular domain in ℂ n , n ∈ ℕ 1 . Of course, only the open discs with the centre = 0 and the radius r > 0, are the bounded complete 1-circular domains G ⊂ ℂ.
In our considerations the Minkowski function G ∶ ℂ n → [0, ∞) 1 3 will be very useful. It is known (see e.g, [26]) that G is a norm in ℂ n if G is a convex bounded complete n-circular domain. This function gives the possibility to redefine the domain G and its boundary G as follows: Now, we recall some information about m-homogeneous polynomials. We say that a function Q m ∶ ℂ n ⟶ ℂ, m ∈ ℕ 1 , is an m-homogeneous polynomial if where L m ∶ (ℂ n ) m ⟶ ℂ is a bounded m-linear function (by Q 0 we note a complex constant). For this reason it is very natural to define (see [4]) the following generalization of the norm of m-homogeneous polynomials Q m ∶ ℂ n → ℂ, i.e., the G -balance of such m-homogeneous polynomials A simple kind of 1-homogeneous polynomials are the linear functionals J, I ∈ (ℂ n ) * of the form Note that for m ∈ ℕ 1 , the mapping is an m-homogeneous polynomial and G (I m ) = 1. By F(G, ℂ m ), m ∈ ℕ 1 , let us denote the space of all functions f ∶ G ⟶ ℂ m , by H G the space of all holomorphic functions f ∈ F(G, ℂ) and by H G (0), H G (1) the collection of all f ∈ H G , normalized by f (0) = 0, f (0) = 1, respectively. Let us recall that every function f ∈ H G has a unique power series expansion for the homogeneous polynomials Q f ,m of functions belonging to the family Note, that the sharpness in the Bavrin's result was understood as follows: There exists a bounded complete 2-circular domain G ⊂ ℂ 2 and a function f ∈ B G which realizes the equality in the above inequality. Let us observe that in general case G⊂ℂ n , n ∈ ℕ 1 , the above inequality takes currently the following form by the definition of the G -balance G (Q m ).
In the paper, we solve for ∈ ℂ, k ∈ ℕ 2 , the problem of the sharp upper estimate for the pairs Q f ,k , Q f ,2k homogeneous polynomials of functions belonging to some Bavrin's subfamilies of the families H G (0), H G (1) (the case k = 1 see the section Final Remarks). Moreover, here the sharpness is understand more generally. It means that for every bounded complete n-circular domain G ⊂ ℂ n there exists a function f, which belongs to a mentioned Bavrin's subfamily and realizes the equality in the above inequality. Note that the afore-mentioned estimate is a generealization of the well known planar Fekete-Szegö [8] result onto the s.c.v. case.
In the sequel we use a special kind of functions symmetry. Let us observe that bounded complete n-circular domains G ⊂ ℂ n are k-symmetric sets, k ∈ ℕ 2 , that is G = G, where = k = exp 2 i k is a generator of the cyclic group of kth roots of unity. For k ∈ ℕ 2 , j ∈ ℤ we define the collections F j,k (G, ℂ m ) of functions f ∈ F(G, ℂ m ), (j, k)-symmetrical, i.e., Now we present a functions decomposition theorem [19].
In the next sections of the paper will be very useful the fact that f 0,k ∈ H G (1) for f ∈ H G (1). Note that 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 ℂ n .
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. Proof Let us recall that the simplest case follows from the well-known inequality For another m, p we use a Krzyż's idea [17,Chapt. 6.2] and some properties of (j, k)-symmetrical functions.
Let us take Then Ψ ∶ U → U and is holomorphic. Thus the (0, m − p)-part Ψ 0,m−p of Ψ transforms U into itself, is holomorphic and Hence, by the Schwarz Lemma (in version with the zero = 0 of multiplicity m − p > 0 ) it fulfils the inequality Therefore, the function maps U into itself, is holomorphic and has the expansion Consequently, replacing m−p ∈ U by ∈ U, we get that the function transforms holomorphically U into itself. Hence, by the first part of the proof, we get the thesis.
Note that the equality in the inequality is attained by the functions F( ) = m , ∈ U and F( ) = p , ∈ U. This completes the proof ◻

Main results
We start this section with an n-dimensional Fekete-Szegö type theorem for bounded holomorphic functions on bounded complete n-circular domains in ℂ n . Note that it is a generalization of a 1-dimensional result given by Keogh and Merkes [14].

Theorem 2.1 Let be a function from the family and has the form
Then, for every k ∈ ℕ 1 and every ∈ ℂ, there holds the inequality The estimate is sharp.
Proof For every arbitrarily fixed z ∈ G we define the function Then Φ is holomorphic, hence Φ ∶ U ⟶ U. Now let us observe that from the Lemma 1.1 we get for arbitrarily fixed m, p ∈ ℕ 1 , satisfying the condition 0 ≤ 2(p − 1) < m − 1. Thus Therefore, for z ∈ G and every ∈ ℂ because Consequently, by the arbitrarinnes of z ∈ G, also Putting in the above p = k and m = 2k, Finally, by the fact that Q ,2k − Q ,k 2 is a 2k-homogeneous polynomial for every ∈ ℂ and by the definition of its G -balance, we get the statements of the Theorem 2.1. Now, we will analyse the sharpness of the above estimate.
In the sequel we apply Theorem 2.1 to study two Bavrin's families M G , N G of functions f ∈ H G (1) . These families are defined by the following family C G , and by the following Temljakov [29] where Df(z) means the Fréchet derivative of f at the point z. Note that the operator L is invertible and It is obvious also that for the transforms L f , L L f of the functions f ∈ H G (1) we have Moreover, We say that a function f ∈ H G (1) belongs to the Bavrin's family M G (N G ) if it satisfies the factorization together with a function h ∈ C G and the transform L f , (L L f ), respectively. Note that the families M G , N G correspond with the well-known families of normalized univalent starlike (convex) functions in the disc U [2] and the family M G can be used to construction biholomorphic starlike mappings in ℂ n (see [6,18], compare also [12,25]). Between functions from M G , N G there holds a relationship, corresponding to the well-known Alexander type connexion [1], for univalent starlike and convex mappings in the unit disc. Here, this relationship is the following: if f ∈ N G , then L f ∈ M G and conversely, if f ∈ M G , then We begin the presentation of some Fekete-Szegö type results in Bavrin's families with the following theorem.  1), with Q f ,0 = 1, then for the homogeneous polynomials Q f ,2k , Q f ,k and every ∈ ℂ there holds the following sharp estimate: Proof Let us recall that between the functions p ∈ C G and ∈ B G (0), there holds the following relation [2]: Let k ∈ ℕ 2 be arbitrarily fixed and let the function f belongs to N G ∩ F 0,k (G,ℂ). Then, by the definition of the family N G and by the above relation between the families C G , B G (0), we get where ∈ B G (0)∩F 0,k (G,ℂ). On the other hand, from (1.1) we have for z ∈ G: and from (2.2) , (2.3) also Inserting the above expansion of functions into (2.7) , we receive after computations Then, comparing the m-homogeneous polynomials of the same degree on both sides of the above equality, we can determine homogeneous polynomials Q ,k , Q ,2k , as follows Putting the above equalities into Theorem 2.1 and using the fact that the mapping Q f ,k 2 , is a 2k-homogenous polynomial, we obtain Indeed. Function f belongs to F 0,k (G,ℂ), because f ∈ F 0,k (G,ℂ). On the other hand, the function f = L −1f belongs to N G , because the function from (2.7) has the form =̃ = I k and belongs to B G (0). Therefore, we can write equalities (2.8 ) in the following form From this, by the case condition for , we have step by step:  1), with Q f ,0 = 1, then for the homogeneous polynomials Q f ,2k , Q f ,k and ∈ ℂ there holds the following sharp estimate: Proof Let k ∈ ℕ 2 be arbitrarily fixed. Then, it is obvious that L −1 f ∈ F 0,k (G,ℂ), because f ∈ F 0,k (G,ℂ). Also the assumption that f ∈ M G , by the relationship of the Alexander type, gives that L −1 f ∈ N G . Hence, we have that L −1 f ∈ N G ∩ F 0,k (G,ℂ). On the other hand, its expansion into a series of m-homogenous polynomials Q f ,m , by ( 2.4), has the form Thus, in view of (2.6) from Theorem 2.2, we get that for every ∈ ℂ Hence, and denoting 2k+1 (k+1) 2 = , finally which is the same as (2.13). It remains to show the sharpness of the estimate (2.13). First, we prove that, in the case the equality in (2.13) is attained by the function f =f defined in (2.9). Of course f ∈ F 0,k (G,ℂ) and, by the Alexander type relationship and the fact that L −1f ∈ N G (see the proof of the sharpness in Theorem 2.2), the function f belongs to M G . Therefore, in view of (2.2) and (2.10), we achieve From this (see the proof of the sharpness in Theorem 2.2) and by the case condition for , we have step by step: Now, we show that, in the case the equality in (2.13) realizes the function f =f defined in (2.11). Of course f ∈ F 0,k (G,ℂ) and, by the Alexander type relationship and the fact that L −1f ∈ N G (see the proof of the sharpness in Theorem 2.2), the function f belongs to M G . Therefore, in view of ( 2.2) and (2.12) we achieve From this (see the proof of the sharpness in Theorem 2.2) and by the case condition for , we conclude that: This completes the proof. ◻ Now, we transfer the statement of Theorem 2.3, onto a family M k G ∩ F 0,k (G,ℂ),k ∈ ℕ 2 . Here, M k G is defined by the factorization similar as for the elements from M G . More precisely, the function f of the right hand side in (2.5) is replaced by a function from F 0,k (G,ℂ), generated by f. Formally, we say that a function f ∈ H G (1) belongs to M k G , k ∈ ℕ 2 , (see [3,5]) if there exists a function h ∈ C G such that where f 0,k ∈ F 0,k (G,ℂ) is the (0, k) -symmetrical component of the function f in the decomposition (1.2) from Theorem A. This family for k = 2 corresponds to a well-known Sakaguchi family [27] of a complex variable functions; strictly speaking of functions univalent starlike with respect to two symmetric points. In the paper [5] it was shown that for k ∈ ℕ

Applications
In this section we apply Theorems 2.3 and 2.4, to obtain a Fekete-Szegö type results for two families of biholomorphic mappings in ℂ n . By S * ( n ) let us denote the family of biholomorphic mappings F ∈ F( n , ℂ n ), F(0) = 0, DF(0) = I onto starlike domains .
F( n ). For a wide collection of references in this area see the monographs [11,16]. In a Kikuchi-Matsuno-Suffridge characterization [15,23,28] of the family S * ( n ) , the collection P( n ) of all holomorphic mappings H ∈ F( n , ℂ n ), H(0) = 0, DH(0) = I, such that Re⟨H(z), z⟩ > 0, z ∈ n �{0}, plays the main role (here ⟨⋅, ⋅⟩ means the Euclidean inner product). This characterization is included in the following theorem: Theorem B A locally biholomorphic mapping F ∈ F( n , ℂ n ) normalized by the conditions F(0) = 0, DF(0) = I, belongs to S * ( n ), iff there exist a mapping H ∈ P( n ) such that Let S * ( n ) be the family of mappings F ∈ S * ( n ) with the factorization where f ∈ H n (1).
In the paper [6] the authors considered a family of biholomorphic mappings S k ( n ), k ∈ ℕ 2 , defined by an equation similar to (3.1). More precisely, the mapping F of the left hand side in (3.1) is replaced by a function from F 1,k ( n , ℂ n ), generated by F.
Formally, we say that a locally biholomorphic mapping F ∈ F( n , ℂ n ), normalized by F(0) = 0, DF(0) = I, belongs to the family S k ( n ), k ∈ ℕ 2 , if it satisfies the equation where H ∈ P( n ) and F 1,k ∈ F 1,k ( n , ℂ n ) is the (1, k)-symmetrical part of F in the decomposition (1.2). Also in the paper [6] the authors proved, that for every k ∈ ℕ 2 any inclusions S k ( n ) ⊂ S * ( n ), S * ( n ) ⊂ S k ( n ) do not holds. However, for the same k there holds the following identity: Let Sk ( n ), k ∈ ℕ 2 , be the family of mappings F ∈ S k ( n ) with the factorization (3.2). Now, we present the main theorem, in this section of the paper. It is a Fekete-Szegö type result for locally biholomorphic mappings in ℂ n , compare [30]. Theorem 3.1 For mappings F ∈S k ( n ) ∩F 1,k ( n , ℂ n ), k ∈ ℕ 2 , the parameter ∈ ℂ and points z ∈ n �{0} there holds the following sharp estimate where T z ∈ (ℂ n ) * , z ∈ n �{0}, is arbitrary functional satisfying the conditions ‖ ‖ T z ‖ ‖ = 1, T z (z) = ‖z‖.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.