Composition, product and sum of analytic functions of bounded L-index in
direction in the unit ball

Bandura, Andriy

doi:10.15330/ms.50.2.115-134

Cited by 7 publications

(6 citation statements)

References 16 publications

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“…Now we consider an application of Theorems 2 and 3. The following proposition can be obtained using similar considerations as in the case of analytic in the unit ball functions of bounded L-index in direction [8].…”

Section: Auxiliary Propositionsmentioning

confidence: 89%

“…Proof. Our proof is similar to proof for analytic in the unit ball functions in [8] but now we use Theorem 2, deduced for functions holomorphic on the slices in the unit ball. Since an analytic function F (z) has bounded L-index in the direction b, by Theorem 2 for every r ∈ (0, β) there exists n(r) ∈ Z + such that for all z 0 ∈ B n , satisfying…”

Section: Auxiliary Propositionsmentioning

confidence: 93%

“…Proof. We repeat our arguments from [8] where this theorem is proved for functions analytic in the unit ball. We write Cauchy's formula for the slice holomorphic function F (z 0 + tb) as analytic function of one complex variable t…”

Section: Auxiliary Propositionsmentioning

confidence: 96%

“…Sum of functions of bounded L-index in direction. Above we wrote that the product of analytic in the unit ball functions of bounded L-index in a direction is a function from the same class ( [8]). But the class of analytic functions of bounded index is not closed under the addition.…”

Section: Auxiliary Propositionsmentioning

confidence: 99%

“…and t 0 = t 0 (z, t) ∈ B(z 0 , t) is chosen in (8) and (1). This means that L(z 0 + tb) ≥ L(z 0 +t 0 b) λ b (1) .…”

Section: Auxiliary Propositionsmentioning

confidence: 99%

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Boundedness of the L-index in a direction of the sum and product of slice holomorphic functions in the unit ball

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 $\mathbb{B}^n=\{z\in\mathbb{C}^{n}: \ |z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}<1\}$ for any $z^0\in\mathbb{B}^n$. For this class of functions there is considered the concept of boundedness of $L$-index in the direction $\mathbf{b},$ where ${L}: \mathbb{B}^n\to\mathbb{R}_+$ is a positive continuous function such that $L(z)>\frac{\beta|\mathbf{b}|}{1-|z|}$ and $\beta>1$ is some constant.There are presented sufficient conditions that the sum of slice holomorphic functions of bounded $L$-index in direction belong this class. This class of slice holomorphic functions is closed under the operation of multiplication.

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Section: Auxiliary Propositionsmentioning

confidence: 89%

Section: Auxiliary Propositionsmentioning

confidence: 93%

Section: Auxiliary Propositionsmentioning

confidence: 96%

Section: Auxiliary Propositionsmentioning

confidence: 99%

“…and t 0 = t 0 (z, t) ∈ B(z 0 , t) is chosen in (8) and (1). This means that L(z 0 + tb) ≥ L(z 0 +t 0 b) λ b (1) .…”

Section: Auxiliary Propositionsmentioning

confidence: 99%

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Boundedness of the L-index in a direction of the sum and product of slice holomorphic functions in the unit ball

2022

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Application of Hayman’s Theorem to Directional Differential Equations With Analytic Solutions in the Unit Ball

Bandura

2024

Stud. Univ. Babes-Bolyai Math.

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In this paper, we investigate analytic solutions of higher order linear non-homogeneous directional differential equations whose coefficients are analytic functions in the unit ball. We use methods of theory of analytic functions in the unit ball having bounded L-index. Our proofs are based on application of inequalities from analog of Hayman’s theorem for analytic functions in the unit ball. There are presented growth estimates of their solutions which contain parameters depending on the coefficients of the equations. Also, we obtained sufficient conditions that every analytic solution of the equation has bounded L-index in the direction. The deduced results are also new in one-dimensional case, i.e. for functions analytic in the unit disc. Keywords: Analytic function, analytic solution, slice function, unit ball, directional differential equation, growth estimate, bounded L-index in direction.

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Note on composition of entire functions and bounded $L$-index in direction

Bandura

Скасків

Сало³

2021

Mat. Stud.

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We study the following question: ``Let $f\colon \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi\colon \mathbb{C}^n\to \mathbb{C}$ an be entire function, $n\geq2,$ $l\colon \mathbb{C}\to \mathbb{R}_+$ be a continuous function. What is a positive continuous function $L\colon \mathbb{C}^n\to \mathbb{R}_+$ and a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ such that the composite function $f(\Phi(z))$ has bounded $L$-index in the direction~$\mathbf{b}$?'' In the present paper, early known result on boundedness of $L$-index in direction for the composition of entire functions $f(\Phi(z))$ is modified. We replace a condition that a directional derivative of the inner function $\Phi$ in a direction $\mathbf{b}$ does not equal zero. The condition is replaced by a construction of greater function $L(z)$ for which $f(\Phi(z))$ has bounded $L$-index in a direction. We relax the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K|\partial_{\mathbf{b}}\Phi(z)|^k$ for all $z\in\mathbb{C}^n$,where $K\geq 1$ is a constant and ${\partial_{\mathbf{b}} F(z)}:=\sum\limits_{j=1}^{n}\!\frac{\partial F(z)}{\partial z_{j}}{b_{j}}, $ $\partial_{\mathbf{b}}^k F(z):=\partial_{\mathbf{b}}\big(\partial_{\mathbf{b}}^{k-1} F(z)\big).$ It is replaced by the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K(l(\Phi(z)))^{1/(N(f,l)+1)}|\partial_{\mathbf{b}}\Phi(z)|^k,$ where $N(f,l)$ is the $l$-index of the function $f.$The described result is an improvement of previous one.

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Composition, product and sum of analytic functions of bounded L-index in direction in the unit ball

Cited by 7 publications

References 16 publications

Boundedness of the L-index in a direction of the sum and product of slice holomorphic functions in the unit ball

Boundedness of the L-index in a direction of the sum and product of slice holomorphic functions in the unit ball

Application of Hayman’s Theorem to Directional Differential Equations With Analytic Solutions in the Unit Ball

Note on composition of entire functions and bounded $L$-index in direction

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