Note on composition of entire functions and bounded $L$-index in direction
We study the following question: ``Let $f\colon \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi\colon \mathbb{C}^n\to \mathbb{C}$ an be entire function, $n\geq2,$ $l\colon \mathbb{C}\to \mathbb{R}_+$ be a continuous function. What is a positive continuous function $L\colo...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | German |
| Published: |
Ivan Franko National University of Lviv
2021-03-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/185 |
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| Summary: | We study the following question: ``Let $f\colon \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi\colon \mathbb{C}^n\to \mathbb{C}$ an be entire function, $n\geq2,$ $l\colon \mathbb{C}\to \mathbb{R}_+$ be a continuous function. What is a positive continuous function $L\colon \mathbb{C}^n\to \mathbb{R}_+$ 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 result on boundedness of $L$-index in direction for the composition of entire functions $f(\Phi(z))$ is modified. We replace a condition that a directional derivative of the inner function $\Phi$ in a direction $\mathbf{b}$ does not equal zero. The condition is replaced by a construction of greater function $L(z)$ for which $f(\Phi(z))$ has bounded $L$-index in a direction. We relax the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K|\partial_{\mathbf{b}}\Phi(z)|^k$ for all $z\in\mathbb{C}^n$,
where $K\geq 1$ is a constant and ${\partial_{\mathbf{b}} F(z)}:=\sum\limits_{j=1}^{n}\!\frac{\partial F(z)}{\partial z_{j}}{b_{j}}, $ $\partial_{\mathbf{b}}^k F(z):=\partial_{\mathbf{b}}\big(\partial_{\mathbf{b}}^{k-1} F(z)\big).$ It is replaced by the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K(l(\Phi(z)))^{1/(N(f,l)+1)}|\partial_{\mathbf{b}}\Phi(z)|^k,$ where $N(f,l)$ is the $l$-index of the function $f.$
The described result is an improvement of previous one. |
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| ISSN: | 1027-4634 2411-0620 |