Majorization of the Temljakov Operators for the Bavrin Families in C n

. The paper concerns holomorphic functions in complete bounded n -circular domains G of the space C n . The object of the present investiga-tion is to solve majorization problem of Temljakov operator. This type of problem has been studied earlier in Liczberski and ˙Zywie´n (Folia Sci Univ Tech Res 33:37–42, 1986), Liczberski (Bull Technol Sci Univ 20:29– 37, 1988) and Le´s-Bomba and Liczberski (Demonstratio Math 42(3):491– 503, 2009). In this paper we considered the family M G ∩ F 0 ,k ( G ), i.e. the functions of the Bavrin family M G , which are (0 , k )-symmetrical.


R. D lugosz et al. Results Math
Moreover, if a domain G is additionally bounded and convex, we have μ G (·) = || · || (see [9]). Let H G denote a family of holomorphic functions f : G → C and let L : H G → H G be the Temljakov linear operator [10], which is defined by where Df (z)(w) is the value of the Frechet's derivative Df (z) of f at the point z on a vector w (here Df (z) is the row vector ∂f (z) ∂z1 , ..., ∂f (z) ∂zn and w is a column vector).
It is also know (see [10]) that the inverse of the Temljakov operator has the following form In order to show our results, we will use the following property of the Temljakov operator. Proof.
After adding and subtracting the product u(z)v(z) in the above equality we obtain We will consider some subfamilies The below subfamilies X G are defined by the family C G , We say that a function f ∈ H G (1) belongs to M G , N G , R G (see [1]) if there exists a function h ∈ C G such that respectively. In the case n = 2 Bavrin ( [1]) gave the following geometrical interpretation for functions from M G . A function f belongs to M G if and only if (i) the function z 1 f (z 1 , z 2 ) is univalent starlike in the intersection of the domain G by every analitic plane z 2 = αz 1 , α ∈ C. In other words the function z 1 f (z 1 , αz 1 ) of one variable is univalent starlike in the disc, which is the projection of the intersection G ∩ {z 2 = αz 1 } onto the plane z 2 = 0, (ii) the function z 2 f (0, z 2 ) is univalent starlike in the intersection G∩{z 1 = 0}.
In connection with this interpretation we say that the family M G corresponds to the class S * of normalize univalent starlike functions F : U → C. In the same way we can say that the family N G (R G ) corresponds to the class S c (S cc ) (see [4]) of normalized univalent convex (close-to-convex) functions.
In the papers [3,5] the notion of G−balance of linear functionals A : C n → C was defined as follows Moreover, if the domain G is also convex then μ G (A) is a norm of the linear functional A.
Therefore μ G ( I) for the linear functional I : C n → C defined by In the sequel I : C n → C is a linear operator defined by We can see that μ G (I) = 1 and we have Let k ≥ 2 be an arbitrarily fixed integer, ε = ε k = exp 2πi k and a set D ⊂ C n be k−symmetric (εD = D). For j = 0, 1, . . . , k−1 we define the spaces A very useful result concerning with (j, k)-symmetrical functions is the following [7]: For every function f : D → C there exists exactly one sequence of func- By the uniqueness of the partition (4) the functions f j,k will be called further (j, k) −symmetrical components of the function f . Moreover, note that n-circular domain is k-symmetric.

The Majorization Problem
Let f, F ∈ H G and r ∈ [0, 1]. If we say that the function F majorizes the function f in the set rG.
The second author (see [6]) has proved that if in a complete bounded two-circular domain G ⊂ C 2 a function F ∈ M G majorizes a function f ∈ H G , then LF majorizes Lf in rG, r = r(M G ) = 2 − √ 3. Moreover, the number r(M G ) cannot be increased by taking G, to be the cone in C 2 A(2; 1) = {z ∈ C 2 : |z 1 | + |z 2 | < 1}.
Moreover, in paper [5] an analogous result optimal in case of any complete bounded n-circular domain G ⊂ C n for the superclass R G of the class M G was given.
The main theorem is preceded by lemma.
Proof. Proof of the relation G z ∈ S * we can find in [1]. The (1, k) symmetry of G z follows from the following equalities Vol. 75 (2020) Majorization of the Temljakov Operators Page 5 of 9 60 Theorem 1. Let G ⊂ C n , n ≥ 2 be a bounded complete n-circular domain. If a function f ∈ H G is majorized in G by a function F ∈ M G ∩ F 0,k , then where and r k is the unique solution in (0, 1) of the equation The function T in (7) cannot be replaced by any function with values T (r) smaller than the values of T defined by (8).

Now we go to the estimate of the expression F (z)
LF (z) . We will use Lemma 2. It is known (see [11]) that for the function G ∈ S * ∩ F 1,k (U) there holds the bound Taking into account the above ξ = μ G (z) = r ∈ [0, 1) we have by (6) 60 Page 6 of 9 R. D lugosz et al.

Results Math
As a result we have Let us consider the right-hand side part of this inequality as a square function of the variable x: Then its maximum in interval [0,1] is equal to and it is 1 for r ∈ [0, r k ]. Indeed, the maximum ordinate y v is attained for the abscissa x v ∈ (0, 1), in opposite case the maximum y v = y v (1) = 1. Setting we have Now, we show that the polynomial q(r) = r k+2 − 2r k+1 − r k − r 2 − 2r + 1 has exactly one root in the interval (0, 1). Finally, it is sufficient to note that q(0)q(1) < 0 and for r ∈ [0, 1) Therefore in the interval (0, 1) the function q(r) has exactly one root r k . For x v given by (11) we have Hence we obtain (8).
In order to prove the second part of the theorem, let us assume that r ∈ [r k , 1), the point where F (z) = 1 I k (z) means the product of k identical factors I(z) (see [2]) and We set the α parameter from the condition , z = (r, 0, . . . , 0), α ∈ (0, 1).
Thus F ∈ M G ∩ F 0,k , f is majorized by F in G and for the point • z we have equality in (7). However, by putting f = F for r ∈ [0, r k ], where F ∈ M G ∩ F 0,k , we have Lf = LF and equality in (7) holds for points z ∈ G such that μ G (z) = r ∈ [0, r k ]. This completes the proof. Corollary 1. Let n ≥ 2 and G be a bounded complete n-circular domain of C n . If a function F ∈ M G ∩ F 0,k majorizes a function f ∈ H G in G, then the function LF majorizes the function Lf in the domain r k G, where r k is the unique solution in (0, 1) of the equation (9). The constant r k cannot be replaced by any greater number r.
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