切片全函数和方向上有界l指数的组合

O. Skaskiv, Andriy Ivanovych Bandura
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引用次数: 1

摘要

我们研究如下问题:设$f: \mathbb{C}\to \mathbb{C}$为有界$l$ -索引的整函数,$\Phi: \mathbb{C}^n\to \mathbb{C}$为切片整函数,$n\geq2,$$l:\mathbb{C}\to \mathbb{R}_+$为连续函数。我们研究如下问题:设$f: \mathbb{C}\to \mathbb{C}$为有界$l$ -索引的整函数,$\Phi: \mathbb{C}^n\to \mathbb{C}$为切片整函数,$n\geq2,$$l:\mathbb{C}\to \mathbb{R}_+$为连续函数。什么是一个正连续函数$L:\mathbb{C}^n\to \mathbb{R}_+$和一个方向$\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$,使得复合函数$f(\Phi(z))$在$\mathbf{b}$方向上有有界的$L$ -指数?本文将先前已知的关于整个函数合成$f(\Phi(z))$的$L$ -方向索引的有界性的结果推广到$\Phi: \mathbb{C}^n\to \mathbb{C}$是一个片整个函数的情况,即对于任意$z^0\in\mathbb{C}^n$和给定方向$\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$,它是复直线$\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$上的一个完整函数。这些片整个函数在一般情况下不是联合全纯的。例如,它允许考虑变量为$z_1$全纯和变量为连续的函数 $z_2.$
本文章由计算机程序翻译,如有差异,请以英文原文为准。
COMPOSITION OF SLICE ENTIRE FUNCTIONS AND BOUNDED L-INDEX IN DIRECTION
We study the following question: "Let $f: \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi: \mathbb{C}^n\to \mathbb{C}$ be a slice entire function, $n\geq2,$ $l:\mathbb{C}\to \mathbb{R}_+$ be a continuous function.We study the following question: "Let $f: \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi: \mathbb{C}^n\to \mathbb{C}$ be a slice entire function, $n\geq2,$ $l:\mathbb{C}\to \mathbb{R}_+$ be a continuous function.What is a  positive continuous function $L:\mathbb{C}^n\to \mathbb{R}_+$ 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 results on boundedness of $L$-index in direction for the composition of entire functions$f(\Phi(z))$ are generalized to the case where  $\Phi: \mathbb{C}^n\to \mathbb{C}$ is a slice entire function, i.e.it is an entire function on a complex line $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ for any $z^0\in\mathbb{C}^n$ andfor a given direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$.These slice entire functions are not joint holomorphic in the general case. For~example, it allows consideration of functions which are holomorphic in variable $z_1$ and  continuous in variable $z_2.$
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