某类积的零点和奇点的间隙条件

Pub Date : 2024-06-22 DOI:10.1007/s11785-024-01564-8
Szymon Ignaciuk, Maciej Parol
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引用次数: 0

摘要

我们对公式$$\begin{aligned}给出的函数的卡普兰类进行完整的成员划分\在{mathbb{C}}:|<1}\ni z\mapsto \prod \limits _{k=1}^n (1-z\textrm{e}^{-\textrm{i}t_k})^{p_k}, \end{aligned}$$其中 \(n\in \mathbb N\),\(t_k\in [0;2pi )和(p_k\in \mathbb R)为(k\in \mathbb N\cap [1;n])。通过这种方式,我们扩展了谢尔-斯莫尔(Sheil-Small)、贾汉吉里(Jahangiri)和我们之前的结果。此外,我们还给出了所获间隙条件的物理和几何应用。首先是质量和密度的解释。第二个是用\(\mathbb {R}^2\) 中矢量之间的角不等式来进行可视化。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

A Gap Condition for the Zeros and Singularities of a Certain Class of Products

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A Gap Condition for the Zeros and Singularities of a Certain Class of Products

We carry out complete membership to Kaplan classes of functions given by formula

$$\begin{aligned} \{\zeta \in {\mathbb {C}}:|\zeta |<1\}\ni z\mapsto \prod \limits _{k=1}^n (1-z\textrm{e}^{-\textrm{i}t_k})^{p_k}, \end{aligned}$$

where \(n\in \mathbb N\), \(t_k\in [0;2\pi )\) and \(p_k\in \mathbb R\) for \(k\in \mathbb N\cap [1;n]\). In this way we extend Sheil-Small’s, Jahangiri’s and our previous results. Moreover, physical and geometric applications of the obtained gap condition are given. The first one is an interpretation in terms of mass and density. The second one is a visualization in terms of angular inequalities between vectors in \(\mathbb {R}^2\).

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