Analytic vector-functions in the unit ball having bounded $\mathbf{L}$-index in joint variables

2020

Mat. Stud.

We consider a class of vector-valued entire functions $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$. For this class of functions there is introduced a concept of boundedness of $\mathbf{L}$-index in joint variables. Let $|\cdot|_p$ be a norm in $\mathbb{C}^p$. Let $\mathbf{L}(z)=(l_{1}(z),\ldots,l_{n}(z))$, where $l_{j}(z)\colon \mathbb{C}^{n}\to \mathbb{R}_+$ is a positive continuous function.An entire vector-valued function $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$ is said to be ofbounded $\mathbf{L}$-index (in joint variables), if there exists $n_{0}\in \mathbb{Z}_{+}$ such that $\displaystyle \forall z\in G \ \ \forall J \in \mathbb{Z}^n_{+}\colon \quad\frac{|F^{(J)}(z)|_p}{J!\mathbf{L}^J(z)}\leq \max \left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^K(z)} \colon K\in \mathbb{Z}^n_{+}, \|K\|\leq n_{0} \right \}.$ We assume the function $\mathbf{L}\colon \mathbb{C}^n\to\mathbb{R}^p_+$ such that $0< \lambda _{1,j}(R)\leq\lambda _{2,j}(R)<\infty$ for any $j\in \{1,2,\ldots, p\}$ and $\forall R\in \mathbb{R}_{+}^{p},$where $\lambda _{1,j}(R)=\inf\limits_{z_{0}\in \mathbb{C}^{p}} \inf \left \{{l_{j}(z)}/{l_{j}(z_{0})}\colon z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \},$ $\lambda _{2,j}(R)$ is defined analogously with replacement $\inf$ by $\sup$.It is proved the following theorem:Let $|A|_p=\max\{|a_j|\colon 1\leq j\leq p\}$ for $A=(a_1,\ldots,a_p)\in\mathbb{C}^p$. An entire vector-valued function $F$ has bounded $\mathbf{L}$-index in joint variables if and only if for every $R\in \mathbb{R}^{n}_+$ there exist $n_{0}\in \mathbb{Z}_{+}$, $p_0>0$ such that for all $z_{0}\in \mathbb{C}^{n}$ there exists $K_{0}\in \mathbb{Z}_{+}^{n}$, $\|K_0\|\leq n_{0}$, satisfying inequality $\displaystyle\!\max\!\left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^{K}(z)} \colon \|K\|\leq n_{0},z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \}%\leq \nonumber\\\label{eq:5}\leq p_{0}\frac{|F^{(K_0)}(z_0)|_p}{K_0!\mathbf{L}^{K_0}(z_0)},$ where $\mathbb{D}^{n}[z_{0},R]=\{z=(z_1,\ldots,z_n)\in \mathbb{C}^{n}\colon |z_1-z_{0,1}|<r_{1},\ldots, |z_n-z_{0,n}|<r_{n}\}$ is the polydisc with $z_0=(z_{0,1},\ldots,z_{0,n}),$\ $R=(r_{1},\ldots,r_{n})$. This theorem is an analog of Fricke's Theorem obtained for entire functions of bounded index of one complex variable.

“…The following theorem is basic in the theory of functions of bounded index. For various classes of analytic functions similar theorems are proved in [1,10,15,17].…”

Section: Notations and Definitionsmentioning

confidence: 85%

“…In recent papers [1,2,3] V. Baksa, A. Bandura, O. Skaskiv considered vector-valued functions having bounded L-index in joint variables which are analytic in the unit ball. They also extended previous investigations of analytic functions in the unit ball ( [6]).…”

mentioning

confidence: 99%

Entire multivariate vector-valued functions of bounded $\mathbf{L}$-index: analog of Fricke’s theorem

2020

Mat. Stud.

“…In this paper, we consider vector-valued functions of bounded L-index in joint variables which are analytic in the unit ball. This paper is a continuation of investigations initiated in [1,2,3]. There was proposed the definition of L-index boundedness in joint variables and obtained some criteria of L-index boundedness in joint variables for vector-valued analytic functions in the unit ball.…”

Section: Introductionmentioning

confidence: 98%

Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded $\mathbf{L}$-index in Joint Variables

Constructive Mathematical Analysis

Скасків

2020

Our results concern growth estimates for vector-valued functions of L-index in joint variables which are analytic in the unit ball. There are deduced analogs of known growth estimates obtained early for functions analytic in the unit ball. Our estimates contain logarithm of sup-norm instead of logarithm modulus of the function. They describe the behavior of logarithm of norm of analytic vector-valued function on a skeleton in a bidisc by behavior of the function L. These estimates are sharp in a general case. The presented results are based on bidisc exhaustion of a unit ball.

“…An analytic vector-function F : B 2 → C 2 is said to be of bounded L-index (in joint variables) [1,2,3], if there exists n 0 ∈ Z + such that…”

Section: Introductionmentioning

confidence: 99%

On Existence of Main Polynomial for Analytic Vector-Valued Functions of Bounded L-Index in the Unit Ball

Скасків

2019

BMJ

In this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-functions analytic in the unit ball, where L = (l 1 , l 2 ) :These conditions describe local behavior of homogeneous polynomials (so-called a main polynomial) with power series expansion for analytic vector-valued functions in the unit ball. These results use a bidisc exhaustion of a unit ball.Key words and phrases: bounded index, bounded L-index in joint variables, analytic function, unit ball, main polynomial, homogeneous polynomial.