Some Characteristic Properties of Analytic Functions in D×ℂ of Bounded L-Index in Joint Variables

Bandura, Andriy; Скасків, О. Б.; Tsvigun, V. L.

doi:10.31861/bmj2018.01.021

Cited by 11 publications

(10 citation statements)

References 9 publications

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“…In recent years, analytic functions of several variables with bounded index have been intensively investigated. The main objects of investigations are such function classes: entire functions of several variables [1][2][3], functions analytic in a polydisc [4], in a ball [5] or in the Cartesian product of the complex plane and the unit disc [6].…”

Section: Introductionmentioning

confidence: 99%

Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

Bandura

Скасків

2019

Axioms

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In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

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Section: Introductionmentioning

confidence: 99%

Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

Bandura

Скасків

2019

Axioms

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“…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: 87%

“…The concept of boundedness of L-index in joint variables were considered for other classes of analytic functions. They are differed in domains of analyticity: the unit ball ( [6]), the polydisc ( [9]), the Cartesian product of the unit disc and complex plane ( [10]), n-dimensional complex space ( [8,11]), slice analyticity ( [5]). By Q n we denote the class of functions L :…”

Section: Notations and Definitionsmentioning

confidence: 99%

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

Bandura

Baksa

2020

Mat. Stud.

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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.

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“…For an analytic function F (z) we put M (R, z 0 , F ) = max{|F (z)| : z ∈ T 2 (z 0 , R)}. The following theorems were obtained in [3].…”

Section: Introductionmentioning

confidence: 99%

Grows estimated of analytic functions in $\mathbb{D}\times\mathbb{C}$ having bounded $\mathbf{L}$-index i joint variables

Bandura¹,

Tsvigun²

2020

vmm

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We prove some properties of power series expansion of analytic functions in D × C having bounded L-index in joint variables, where L(z, w) = (l1(z, w), l2(z, w)) with lj : D × C → R+ (j ∈ {1, 2}) are positive continuous functions and l1(z, w) > β/(1 − |z|) for all (z, w) ∈ D × C and some β > 1. Moreover, we provide growth estimates of these function class. They describe the behavior of logarithm of maximum modulus of analytic 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 the Cartesian product of the unit disc and complex plane.

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Some Characteristic Properties of Analytic Functions in D×ℂ of Bounded L-Index in Joint Variables

Cited by 11 publications

References 9 publications

Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

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

Grows estimated of analytic functions in $\mathbb{D}\times\mathbb{C}$ having bounded $\mathbf{L}$-index i joint variables

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