Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of 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}

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families CG,MG,NG,RG,VG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}$$\end{document} of such holomorphic functions on complete n-circular domain G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}$$\end{document} of 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 some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families KGk,k≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,$$\end{document} of holomorphic functions f : G→C,f(0)=1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}\rightarrow \mathbb {C},f(0)=1,$$\end{document} defined also by a factorization of Lf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal {L}f$$\end{document} onto factors from CG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{\mathcal {G}}$$\end{document} and MG.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} _{\mathcal {G}}.$$\end{document} We present some interesting properties and extremal problems on KGk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_{\mathcal {G}}^{k}$$\end{document}.


Introduction
We say that a domain G ⊂ C n , is complete n-circular if zλ = (z 1 λ 1 , . . . , z n λ n ) ∈ G for each z = (z 1 , . . . , z n ) ∈ G and every λ = (λ 1 , . . . , λ n ) ∈ U n , where U is the unit disc {ζ ∈ C : |ζ | < 1}. From now by G will be denoted a bounded complete n-circular domain in C n , n ≥ 2. By H G let us denote the space of all holomorphic functions f : G −→ C and by H G (1) the collection of all f ∈ H G , normalized by f (0) = 1.
Many authors (cf., eg., [1,2,[5][6][7]11,18,19,23]) considered some Bavrin's subfamilies X G of the family H G (1). In the definitions of these families X G the main role play the families C G (α), α ∈ [0, 1), and the following invertible Temljakov [24] linear operator L : where D f (z) is the Fréchet derivative of f at the point z. By a Bavrin's family X G we mean a collection of functions f ∈ H G (1) whose the Temljakov transform L f has a functional factorization L f = p · g, where p ∈ C G ≡ C G (0) and g is from a fixed subfamily of H G (1). Below, we recall the factorizations which define a few well known Bavrin's families X G , like as It is known that functions of these families were used to construct biholomorphic mappings in C n (cf., eg., [10,13,20]). Let us note that the above families have geometric interpretation, in particular the functions f ∈ M G map biholomorphically some planar intersections S of G onto starlike domains in C,(see [1]). It is very important, because the starlikeness plays a central role in many different subjects of geometry and topology and in particular, in geometric function theory.
Let us recall also that Bavrin showed the inclusions N G R G , V G R G and pointed that the first of them can be complete to the following double inclusion N G M G R G . Thus, it is natural to ask whether is possible to do the same in the case of the second above inclusion. In the paper [12] the authors defined a family K − G , which satisfies the inclusion where the family M G (α), α ∈ [0, 1), is defined similarly as M G , but in this case p ∈ C G (α).
In the present paper we consider Bavrin's type families The formal definition of such family has the following form.
where ε = ε k = exp 2πi k is a generator of the cyclic group of kth roots of unity. Let us observe that K k G , k ≥ 2 are nonempty families. Indeed, the function f = 1 belongs to K k G , because it satisfies the factorization (1.1) with p = 1 ∈ C G and h = 1 ∈ M G ( k−1 k) ). In the future, we will use a characterization of the family K k G by a notion of ( j, k)symmetry, which is connected with a functional decomposition with respect to the above group.
Let us observe that bounded complete n-circular domains G are k-symmetric sets for k = 2, 3, . . . , that is εG = G. For j = 0, 1, . . . , k − 1 we define the collections F j,k (G) of functions ( j, k)-symmetrical, i.e., all functions f : G → C such that If n = 1 and G = U, then we write F j,k (U).
The mentioned functional decomposition appears in the following result from [14].
Theorem A For every function f : G → C there exists exactly one sequence of func- Moreover, The functions f j,k , which are uniquely determined by the above decomposition, will be called ( j, k)-symmetrical components of the function f. Some interesting applications of the above partition may also be found in [15,16] and [17].

Results
Now we can present a characterization of f ∈ K k G , simpler than (1.1).
Theorem 1 A function f ∈ H G (1) belongs to the family K k G , k ≥ 2 if and only if there exists a function g ∈ M G ∩ F 0,k (G) and a function p ∈ C G such that It is obvious that g ∈ F 0,k (G). We show that g ∈ M G . To do it, using the differentiation product rule and the form of the operator L, we have at z ∈ G Lg(z) Hence and by the fact Now, let us suppose that f satisfies the equality (2.1), with a p ∈ C G and a g ∈ M G ∩ F 0,k (G). Let us put h(z) = (g(z)) 1 k , z ∈ G, with the power function taking the value 1 at the point 1. Since g(z) = 0 (see [1]), the function h is holomorphic. It remains to show that h ∈ M G ( k−1 k ) and the equality (1.1) is fulfilled. To this end we compute step by step The formula (1.1) follows from the definition of the function h. Indeed, The proof is complete. Now we consider an extremal problem for f ∈ K k G . More precisely, we look for some estimates for G-balances of m -homogeneous polynomias Q f,m of its unique power series expansion In our considerations the Minkowski function will be very useful. This function gives a possibility to redefine the domain G and its boundary ∂G as follows: The notion of G-balance of m-homogeneous polynomial Q m : C n → C, m ∈ N∪{0}, was defined in [3] as the quantity and for bounded convex complete n-circular domains G also μ G (w) = ||w|| (see, e.g., [21]). We present the announced estimates of G-balances μ G (Q f,m ) of m-homogeneous polynomials Q f,m from the Taylor series of f ∈ M k G in the following theorem.

Theorem 2 If the expansion of the function f
of polynomials Q f,m the following sharp estimate hold: where q means the integral part of the number q. We use a standard convention that the product l 2 l=l 1 a l is equal to 1 for l 2 < l 1 .
Proof Let f ∈ K k G be arbitrarily fixed. Then, by Theorem 1, the factorization (2.1) holds with a function p ∈ C G of the form From the above, by the series expansion of L f (see [1]) we need some bounds for Q g,kμ (z) . We show that for g ∈ M G ∩ F 0,k (G) and μ ∈ N there hold the inequalities For this purpose let us observe that for each z ∈ G, the function belongs to the family S * ∩ F 1,k (U) of (1, k)-symmetric univalent starlike mappings (in the unit disc U) and its Taylor series has the form Thus, in view of the estimates [25] of the coefficients of functions from S * ∩ F 1,k (U) we get the announced bounds (2.6).
In two next parts of the proof we use also the fact [4] that for every k, s ∈ N\ {1} there holds the identity: Now, we will estimate the quantities Q f,m (z) , z ∈ G, using all the conditions (2.4)-(2.7). First let us assume that m = ks, where s ∈ N. Since Q p,m−kl (z) = 1 for l = s, we get from (2.4) that Thus for z ∈ G, in view of (2.6) and (2.7), Hence, for m = k, 2k, 3k, . . .

defined by Pochhamer symbols (a) ν , (b) ν , (c) ν :
(a) ν = a(a + 1) . . . (a + ν − 1), ν ∈ N 1, ν = 0 , and the branch of the power function x 2 k takes the value 1 at the point x = 1. In the case k = 2, 3 we use a standard convention that the sum is equal to zero, if the superscript of the sum is smaller than the subscript. In the paper [4], it was proven that the above function gives the equalities in the bounds from the statement of the theorem. It remains to show that f ∈ K k G for k ≥ 2. To do it, let us observe that as shown in [4] L f (z) This implies, in view of Theorem 1, the relation f ∈ K k G , because the functions belong to C G and to M G ∩ F 0,k (G), respectively.
We use the estimates of G-balances μ G (Q f,m ) of polynomials Q f,m to solve the mentioned separation problem for the families V G , K k G , R G . We prove the following theorem:

Theorem 3 For every k ≥ 2 there holds the double inclusion
Putting p = L f and h = 1, we obtain the factorization (1.1) with p ∈ C G and g = 1 ∈ M G ∩F 0,k (G). Hence f ∈ K k G . It remains to show the relation V G = K k G . To do it, let us observe that for f ∈ V G there hold the sharp estimates μ G (Q f,m ) ≤ 2 m+1 , m ∈ N (cf., eg., [1]), while for f ∈ K k G the sharp estimates μ G (Q f,m ) ≤ B(m) (Theorem 2.), with the obvious bound B(m) > 2 m+1 , m ∈ N {1}. Hence, the extremal function f ∈ K k G does not belong to V G . Now we prove that K k G ⊂ R G .To this end, let us suppose that f ∈ K k G . Then there exist functions p ∈ C G , g ∈ M G ∩F 0,k (G) such that L f = p·g. Denoting ϕ = L −1 g, we have that ϕ ∈ N G (by the Aleksander type theorem [1]) and L f = pLϕ. Thus f ∈ R G . It remains to show the relation K k G = R G . For this purpose, let us observe that in the above estimates μ G (Q f,m ) ≤ B(m), m ∈ N, we have B(m) ≤ 1, m ∈ N (see below), while for f ∈ R G there hold the sharp estimates μ G (Q f,m ) ≤ m + 1(see for instance [1]). Therefore, the extremal function f ∈ R G does not belong to K k G . To complete the proof, we show that B(m) ≤ 1, m ∈ N. To do it, we consider two cases, according to the partition m = ks + r, r ∈ {0, 1, . . . , k − 1}, from the proof of Theorem 2.
1. Let us suppose that r = 0. Then, if s = m k = 1, we see that the superscript s − 1 of the first product in Theorem 2 is smaller than its subscript 1. Hence, we replace the referred product by 1 and consequently, we get μ G (Q f,m ) ≤ 2 m ≤ 1, because m = k ≥ 2. Next, if s ≥ 2, then from Theorem 2, by the inequality 1

2.
Let us suppose that r ∈ {1, . . . , k − 1}. Then, if s = m k = 0, we see that the superscript of the second product in Theorem 2 is smaller than its subscript. Hence we replace the referred product by 1 and consequently, we get μ G (Q f,m ) ≤ 2 m+1 ≤ 1, because m ≤ k − 1. Next, if s = m k ≥ 1, then similarly as in step 1, we obtain Now, we give a growth theorem for f ∈ K k G and its Temljakov transform L f.
Theorem 4 For functions f ∈ K k G there follow the following sharp estimates Proof First, let us observe that the above estimates are true for z = 0 (in (2.10) the values at r = 0, of the left and right hand sides, mean the limit if r → 0 + ). Thus, in the sequel we will assume that z ∈ G {0}. We start with the estimates (2.9). Since f ∈ K k G , there exist a function p ∈ C G and a function g ∈ M G ∩ F 0,k (G) such that the factorization (2.1) holds. Therefore, we show for such functions g the following inequalities To this aim, let us fix arbitrarily a point z ∈ G such that μ G (z) = r ∈ (0, 1) and let us consider the function Then G is (1, k)-symmetric, holomorphic, normalized and satisfies the condition Hence G ∈ S * ∩ F 1,k (U) and by [9,Thm. 2.2.13] |ζ | Putting ζ = μ G (z) in the above we obtain, by the definition of the function G, the announced inequality.
On the other hand, there hold for p ∈ C G the following estimates [1] Using the estimates of | p(z)| and |g(z)| we get the estimates (2.9). The sharpness of the upper bounds (2.9) confirms the function given by (2.8). Indeed, for r ∈ (0, 1) and function f ∈ K k G given by (2.8), we get at points z ∈ G, μ G (z) = r ∈ (0, 1) such that I (z) = r (this condition is fulfilled by the points z = r z * , where z * ∈ ∂G and I (z * ) = 1). The sharpness of the lower bounds (2.9) can be proven in a similar way. Now, we prove the estimates (2.10). To obtain the upper bound (2.10), we use the proved above upper bound (2.9) and the fact that the Temljakov operator L is invertible and Indeed, we have for f ∈ K k G and z ∈ G, μ G (z) = r ∈ (0, 1), To prove the lower bound (2.10) let us consider the function with arbitrarily fixed f ∈ K k G and z ∈ G, μ G (z) = r ∈ (0, 1). Since we get, by Theorem 1, that there exist functions g ∈ M G ∩ F 0,k (G) and p ∈ C G such that the factorization (2.1) is true. Thus where for ζ ∈ U Moreover, G ∈ S * ∩ F 1,k (U) ( see the proof of the estimates (2.9)) and P : U → C, P(0) = 1, is a holomorphic function with a positive real part. Therefore, F belongs to a subclass K (k) (considered in [22] and for k = 2 in [8]) of the class of close-to-convex functions. Hence, F is univalent in the disc U.
On the other hand, by the lower bound (2.9), we have that To this aim, it is sufficient to show that it holds for the nearest point F(ζ 0 ) from zero (|ζ 0 | = r ∈ (0, 1)), otherwise, we have |F(ζ )| ≥ |F(ζ 0 )|, |ζ | = r. Since F is univalent in the disc U, the original image of the line segment 0, F(ζ 0 ) is a piece of arc F −1 0, F(ζ 0 ) in the disc r U. Thus Thus, by the definition of F, we get Hence, putting ζ = μ G (z) = r ∈ (0, 1), we have the lower bound (2.10).
Finally, let us note that we obtain the equalities in the inequalities (2.10) for the function (2.8) in adequate points z ∈ G.
We close the paper with a sufficient condition guaranteeing that a function f ∈ H G (1) belongs to K k G . We formulate it in the term of G-balances of m-honogeous polynomials in developments of functions from H G (1).
Theorem 5 Let f ∈ H G (1) has the form (2.2). If there exists a function g ∈ M G ∩ F 0,k (G) of the form (2.3) such that ∞ m=1 (m + 1)μ G Q f,m + ∞ m=1 μ G Q g,mk ≤ 1, Proof Since g, as a function from M G omits zero [1], we will prove that To do it, we compute step by step This gives the mentioned inequality by a maximum principle for pluriharmonic functions of several complex variables. Putting p(z) = L f (z) g(z) , z ∈ G, we obtain that the transform L f has the factorization (1) with g ∈ M G ∩ F 0,k (G) and p ∈ C G . Consequently, f ∈ K k G .