Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

Baksa, V. P.; Bandura, Andriy; Скасків, О. Б.

doi:10.1515/ms-2017-0420

Cited by 3 publications

(3 citation statements)

References 22 publications

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“…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: 95%

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

Baksa

Bandura

Скасків

2020

Constructive Mathematical Analysis

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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.

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Section: Introductionmentioning

confidence: 95%

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

Baksa

Bandura

Скасків

2020

Constructive Mathematical Analysis

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“…Nevertheless some important assertions have not full analogs for the bounded L-index in joint variables. For example, in the case of the logatithmic criterion [6,13,27] we know only sufficient conditions for the bounded L-index in joint variables [4,8]. The notion of L-index in direction is more flexible and admits more direct generalizations.…”

Section: Introductionmentioning

confidence: 99%

Analytic in the unit polydisc functions of bounded L-index in direction

Bandura,

Salo

2023

Mat. Stud.

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The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.

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“…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

Bandura

Baksa

Скасків

2019

BMJ

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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.

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Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

Cited by 3 publications

References 22 publications

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

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

Analytic in the unit polydisc functions of bounded L-index in direction

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

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