Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties
| Autor: | A. I. Bandura, T. M. Salo, O. B. Skaskiv |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Matematychni Studii. 57:68-78 |
| ISSN: | 2411-0620 1027-4634 |
| DOI: | 10.30970/ms.57.1.68-78 |
| Popis: | 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 $\mathbb{B}^n=\{z\in\mathbb{C}^: \ |z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}\frac{\beta|\mathbf{b}|}{1-|z|}$ 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. |
| Databáze: | OpenAIRE |
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