Vector-Valued Entire Functions of Several Variables: Some Local Properties

Bandura, Andriy; Сало, Т. М.; Скасків, О. Б.

doi:10.3390/axioms11010031

Cited by 2 publications

(1 citation statement)

References 14 publications

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“…Actually, in Proposition (1) and Theorem 4, the vector-valued entire functions having bounded index in joint variables have implicitly appeared. There are few papers on this class of functions of a single variable [27][28][29], and of several variables [26,30].…”

Section: Examplementioning

confidence: 99%

L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

2023

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The composition H(z)=f(Φ(z)) is studied, where f is an entire function of a single complex variable and Φ is an entire function of n complex variables with a vanished gradient. Conditions are presented by the function Φ providing boundedness of the L-index in joint variables for the function H, if the function f has bounded l-index for some positive continuous function l and L(z)=l(Φ(z))(max{1,|Φz1′(z)|},…,max{1,|Φzn′(z)|}),z∈Cn. Such a constrained function L allows us to consider a function Φ with a nonempty zero set. The obtained results complement earlier published results with Φ(z)≠0. Also, we study a more general composition H(w)=G(Φ(w)), where G:Cn→C is an entire function of the bounded L-index in joint variables, Φ:Cm→Cn is a vector-valued entire function, and L:Cn→R+n is a continuous function. If the L-index of the function G equals zero, then we construct a function L˜:Cm→R+m such that the function H has bounded L˜-index in the joint variables z1,…,zn. The other group of our results concern a sum of entire functions in several variables. As a general case, a sum of functions with bounded index is not of bounded index. The same is also valid for the multidimensional case. We found simple conditions proving that f1(z1)+f2(z2) belongs to the class of functions having bounded index in joint variables z1,z2. We formulate some open problems based on the deduced results and on the usage of fractional differentiation operators in the theory of functions with bounded index.

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Section: Examplementioning

confidence: 99%

L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

2023

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Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties

2022

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Let $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ with the unit ball $\mathbb{B}^n=\{z\in\mathbb{C}^: \ |z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}<1\}$ for any $z^0\in\mathbb{B}^n$. For this class of functions we consider the concept of boundedness of $L$-index in the direction $\mathbf{b},$ where $\mathbf{L}: \mathbb{B}^n\to\mathbb{R}_+$ is a positive continuous function such that $L(z)>\frac{\beta|\mathbf{b}|}{1-|z|}$ and $\beta>1$ is some constant.For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications todifferential equations. We introduce a concept of function having bounded value $L$-distribution in direction forthe slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value $L$-distribution in a direction if and only if its directional derivative has bounded $L$-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function $F$ with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function $L$that the function $F$ has bounded $L$-index in the direction.

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Vector-Valued Entire Functions of Several Variables: Some Local Properties

Cited by 2 publications

References 14 publications

L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties

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