Abstract
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.
Highlights
We assume that G is a bounded complete n-circular domain
Remember that μG is a seminorm in Cn for complete n-circular domain G and is a norm in Cn in the case if G is convex
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
Summary
Remember that μG is a seminorm in Cn for complete n-circular domain G and is a norm in Cn in the case if G is convex Taking this fact into account, we will use a generalization μG ( Q f ,k ) of the norm of k-homogeneous polynomials Q f ,k (see [1]). The family KG (γ) corresponds to the class Ks (γ) of functions of one complex variable introduced by Kowalczyk and Leś-Bomba (see [8]) defined as follows. While presenting the properties of the family KG (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]), n.
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