Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties
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 $\m...
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Ivan Franko National University of Lviv
2022-03-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/311 |
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| author | A. I. Bandura T. M. Salo O. B. Skaskiv |
| author_facet | A. I. Bandura T. M. Salo O. B. Skaskiv |
| author_sort | A. I. Bandura |
| collection | DOAJ |
| description | 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 to
differential equations. We introduce a concept of function having bounded value $L$-distribution in direction for
the 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. |
| format | Article |
| id | doaj-art-dea15d44db984903a4459aaa7d42f44b |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2022-03-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-dea15d44db984903a4459aaa7d42f44b2025-08-20T03:28:41ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202022-03-01571687810.30970/ms.57.1.68-78311Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related propertiesA. I. Bandura0T. M. Salo1O. B. Skaskiv2Ivano-Frankivsk National Technical University of Oil and GasLviv Politechnic National UniversityIvan Franko National University of Lviv, Lviv, UkraineLet $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ 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 to differential equations. We introduce a concept of function having bounded value $L$-distribution in direction for the 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.http://matstud.org.ua/ojs/index.php/matstud/article/view/311bounded index; bounded l-index in direction; slice function; holomorphic function; maximum modulus; minimum modulus; bounded l-index; existence theorem; distribution of zeros; unit ball. |
| spellingShingle | A. I. Bandura T. M. Salo O. B. Skaskiv Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties Математичні Студії bounded index; bounded l-index in direction; slice function; holomorphic function; maximum modulus; minimum modulus; bounded l-index; existence theorem; distribution of zeros; unit ball. |
| title | Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties |
| title_full | Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties |
| title_fullStr | Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties |
| title_full_unstemmed | Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties |
| title_short | Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties |
| title_sort | slice holomorphic functions in the unit ball boundedness of l index in a direction and related properties |
| topic | bounded index; bounded l-index in direction; slice function; holomorphic function; maximum modulus; minimum modulus; bounded l-index; existence theorem; distribution of zeros; unit ball. |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/311 |
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