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

Q3 Mathematics
Andriy Ivanovych Bandura, T. Salo, O. Skaskiv
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引用次数: 0

Abstract

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.
单位球上的切片全纯函数:L -指标在一个方向上的有界性及相关性质
设$\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$为固定方向。我们考虑单位球中几个复变量的切片全纯函数,即我们研究在每个切片$\{z^0+t\mathbf{b}:t\in\mathbb{C}\}$与单位球$\mathbb{b}^n=\{z \ in\mathbb{C}^:\|z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}\frac{β|\mathbf{b}{1-|z|}$的交集上解析的函数,并且$\beta>1$是一些常数。对于这一类的函数,我们推导出海曼定理的类似式。它在微分方程的应用中是有用的判据。对于单位球中的切片全纯函数,我们引入了在方向上具有有界值$L$-分布的函数的概念。证明了单位球中的切片全纯函数在一个方向上具有有界值$L$-分布,当且仅当其方向导数在同一方向上有界$L$。其他命题涉及存在性定理。我们证明了对于在固定方向上的任何切片上具有有界零乘性的任何切片全纯函数$F$,存在这样一个正连续函数$L$,使得函数$F美元在该方向上具有有边界的$L$-索引。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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