Extremal Problems of Some Family of Holomorphic Functions of Several Complex Variables

Many authors, e.g., Bavrin, Jakubowski, Liczberski, Pfaltzgraff, Sitarski, Suffridge, and Stankiewicz, have discussed some families of holomorphic functions of several complex variables described by some geometrical or analytical conditions. We consider a family of holomorphic functions of several complex variables described in n-circular domain of the space C n . We investigate relations between this family and some of type of Bavrin’s families. We give estimates of G-balance of k-homogeneous polynomial, a distortion type theorem and a sufficient condition for functions belonging to this family. Furthermore, we present some examples of functions from the considered class.

We shall use the continuity of µ G and the following facts as well: Remember that µ G is a seminorm in C n for complete n-circular domain G and is a norm in C n in the case if G is also convex. Taking this fact into account, we will use a generalization µ G (Q f ,k ) of the norm of k-homogeneous polynomials Q f ,k (see [1]). In view of the k-homogeneity of Q f ,k , the formula for ∂G and the maximum principle for modulus of holomorphic functions of several variables, we can put for k ∈ N µ G (Q f ,k ) = sup For every k ∈ N, the quantity µ G (Q f ,k ) has the following basic property where D f (z) is the Frechet derivative of f at the point z.
Bavrin (see [4,5]) considered the subclasses V G (γ), M G (γ), N G , and R G of the class We say that f ∈ H G (1) belongs to • R G , if there exists a function φ ∈ N G such that In particular, V G (0) = V G and M G (0) = M G . In the case n = 2, Bavrin (see [4]) gave the following geometrical interpretation for functions from M G . A function f ∈ H G (1) belongs to M G if two conditions are maintained: (i) The function z 1 f (z 1 , αz 1 ) of one variable z 1 is starlike univalent in the disc, which is the projection onto the plane z 2 = 0 of the intersection of the domain G, and every analytic plane (z 1 , z 2 ) ∈ C 2 : z 2 = αz 1 , α ∈ C;. (ii) The function z 2 f (0, z 2 ) is starlike univalent on the intersection G and the plane z 1 = 0.. In connection with this interpretation, we may say that the family M G corresponds to the well-known class S * of the normalized univalent starlike functions F : E → C. In the same way, we can say that the family N G (R G ) corresponds to the class S c (S cc ) (see [6]) of normalized holomorphic univalent convex (close-to-convex) functions.
Note that the class M G has been used in research of some linear invariant families of locally biholomorphic mappings in C n (see [7]).
Here, we consider a subfamily K − G (γ) of the family H G (1). We say that f ∈ H G (1) belongs to K − G (γ), 0 ≤ γ < 1, if there exists such a function, h ∈ M G ( 1 2 ), that satisfies the condition The family K − G (γ) corresponds to the class K s (γ) of functions of one complex variable introduced by Kowalczyk and Leś-Bomba (see [8]) defined as follows.
Let F : E → C, F(0) = 0, F (0) = 1, be a holomorphic function in E. We say that F ∈ K s (γ), The family K − G (γ) is in a way associated with the family K − G considered by Leś-Bomba and Liczberski in [1]. In particular when γ = 0, we have K − While presenting the properties of the family K − G (γ), we will use the number ∆(G)-characteristic, which is assigned to each bounded complete n-circular domain, G, by the following formula (see [4]), Now, we present two examples of functions from this family: where log 1 = 0 and √ 1 = 1. Then, the function f 1 belongs to the family K − G (γ).
Indeed, this function belongs to H G (1) (has a holomorphic extension defined on G, A is a nowheredense and closed subset of G (see [9])) and f 1 for z ∈ G \ A expands into a series of homogenous polynomials Moreover, Let us consider the function h 1 ∈ M G ( 1 2 ) (see [1]) of the form Therefore the condition (7) holds and f 1 ∈ K − G (γ).
Indeed, this function belongs to H G (1) and for z ∈ G \ A it expands into a series of homogenous polynomials e iα ∑ n j=1 z j ∆ 2 + (higher order terms). Moreover, Let us consider the function h 2 ∈ M G ( 1 2 ) (see [1]) of the form Therefore the condition (7) holds and f 2 ∈ K − G (γ).

Main results
The relation between the class K − G (γ) and another type of Bavrin's families is the following. Theorem 1. Let 0 ≤ γ < 1 and let G ⊂ C n be a bounded complete n-circular domain. Then, the following inclusions hold, Proof. Firstly, we show that V G (γ) ⊂ K − G (γ). Let us assume that f ∈ V G (γ). Then condition (3) is satisfied. Let us note that the function h = 1 belongs to the family M G ( 1 2 ) and This means that f belongs to the family K − G (γ). Now, we show that V G (γ) = K − G (γ). Let us consider the function f 2 ∈ K − G (γ) (see Example 2) with α = 0. Then there exists a point, . Therefore, for 0 ≤ γ < 1 the condition (7) occurs, but since γ ≥ 0 the condition (9) also applies. Thus f ∈ K − G .
In the paper, [1] it has been proved that K − G R G , so we have Corollary 1. Let 0 ≤ γ < 1 and let G ⊂ C n be a bounded complete n-circular domain. Then the following inclusion holds We will now present estimates of G-balance of k-homogenous polynomial Q f ,k in the family K − G (γ).

Theorem 2.
Let 0 ≤ γ < 1 and let G ⊂ C n be a bounded complete n-circular domain and let f ∈ K − G (γ). If the expansion of the holomorphic function f into a series of homogeneous polynomials is of the form The estimates are sharp.
(15) Assuming that and therefore, in view of the fact that and the uniqueness theorems for expansions into series of homogenous polynomials, we obtain for k ∈ N, We have (see [10]) and we have (see [1]) because Q g,2m−1 = 0 for m ∈ N (g in Equation (15) is even). While, identity (19) implies that for z ∈ G and k ∈ N, we obtain f or k even, Having (1), we obtain (14). It remains to show the sharpness of (14). Let us observe that the function f 2 ∈ K − G (γ) of the form (12) is the extremal function. Indeed, as the homogenous polynomials Q f ,k in its development (13), k ∈ N and 0 ≤ γ < 1 have the form Moreover, if k ∈ N is even, using (10), we have In the same way we get for a natural odd k that Now, we prove a sufficient condition for functions belonging to the investigated class K − G (γ).

Theorem 3.
Let h ∈ M G ( 1 2 ), 0 ≤ γ < 1 and g(z) = h(z)h(−z) for z ∈ G expands, as in (17). If the expansion of the holomorphic function, f , into a series of homogeneous polynomials is of the form (13) and the function f satisfies the condition then f belongs to K − G (γ) and it is generated by h.
Proof. Let 0 ≤ γ < 1. If the expansion of f into a series of homogenous polynomials has the form (13), then L f (z) has the form (18). Let Therefore, for z ∈ G from (24), we have the inequalities Thus, we obtain which is equivalent to the inequality and consequently we have (7). Thus, f ∈ K − G (γ), which completes the proof.
Below, we provide a distortion type theorem for the considered family of functions.
Theorem 4. Let 0 ≤ γ < 1 and 0 ≤ r < 1, and let G ⊂ C n be a bounded complete n-circular domain. If f ∈ K − G (γ), then The both lower estimates and the upper estimation in (27) are sharp.
Proof. Let 0 ≤ γ < 1. In the case r = 0, the estimates (26) and (27) hold (in (27) for r = 0; it is understood as a limit when r tends to 0). Let r ∈ (0, 1) and µ G (z) = r. First, we put where h ∈ M G ( 1 and | f 1 ( which makes the lower estimations (26) and (27) sharp for z ∈ G, such that µ G (z) ≤ r, i.e., for z ∈ rG, r ∈ (0, 1). Similarly, we can show that the function f 2 ∈ K − G (γ) gives the equality in the upper part of the inequality (27). The upper estimation in (26) is not sharp.