Composition of entire function and analytic functions in the unit ball with a vanished gradient
The composition $H(z)=f(\Phi(z))$ is studied, where $f$ is an entire function of a single complex variable and $\Phi$ is an analytic function in the $n$-dimensional unit ball with a vanished gradient. We found conditions by the function $\Phi$ providing boundedness of the $\mathbf{L}$-index in join...
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Ivan Franko National University of Lviv
2024-12-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/578 |
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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 | The composition $H(z)=f(\Phi(z))$ is studied,
where $f$ is an entire function of a single complex variable and $\Phi$ is an analytic function
in the $n$-dimensional unit ball with a vanished gradient.
We found conditions by the function $\Phi$ providing boundedness of the $\mathbf{L}$-index in joint variables for the function $H$, if the function $f$ has bounded $l$-index for some positive continuous function $l$
and $\mathbf{L}(z)= l(\Phi(z))(\max\{1,|\Phi_{z_1}'(z)|\},\ldots, \max\{1,|\Phi_{z_n}'(z)|\}),$ $z\in\mathbb{B}^n.$
Such a constructed function $\mathbf{L}$ allows us to consider a function $\Phi$ with a nonempty zero set for its gradient.
The obtained results complement earlier published results with $\mathop{grad}\Phi(z)=(\frac{\partial \Phi(z)}{\partial z_1}, \ldots, \frac{\partial \Phi(z)}{\partial z_j},\ldots,\frac{\partial \Phi(z)}{\partial z_n})\ne \mathbf{0}.$
Also, we study a more general composition $H(\mathbf{w})=G(\mathbf{\Phi}(\mathbf{w}))$, where
$G: \mathbb{C}^n\to \mathbb{C}$ is an entire function of the bounded $\mathbf{L}$-index in joint variables,
$\mathbf{\Phi}: \mathbb{B}^m\to \mathbb{C}^n$ is a vector-valued analytic function, and
$\mathbf{L}: \mathbb{C}^n\to\mathbb{R}^n_+$ is a continuous function.
If the $\mathbf{L}$-index of the function $G$ equals zero, then we construct a function
$\widetilde{\mathbf{L}}: \mathbb{B}^m\to\mathbb{R}^m_+$ such that the function $H$ has bounded
$\widetilde{\mathbf{L}}$-index in the joint variables $w_1,$ $\ldots,$ $w_m$.
These results are also new in one-dimensional case, i.e. for functions analytic in the unit disc. |
| format | Article |
| id | doaj-art-6f2f09b2fd544cb8870bfa8bf83f72fe |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2024-12-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-6f2f09b2fd544cb8870bfa8bf83f72fe2025-08-20T03:33:27ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-12-0162213214010.30970/ms.62.2.132-140578Composition of entire function and analytic functions in the unit ball with a vanished gradientA. I. Bandura0T. M. Salo1O. B. Skaskiv2Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, UkraineLviv Politechnic National University, Lviv, UkraineIvan Franko National University of Lviv, Lviv, UkraineThe composition $H(z)=f(\Phi(z))$ is studied, where $f$ is an entire function of a single complex variable and $\Phi$ is an analytic function in the $n$-dimensional unit ball with a vanished gradient. We found conditions by the function $\Phi$ providing boundedness of the $\mathbf{L}$-index in joint variables for the function $H$, if the function $f$ has bounded $l$-index for some positive continuous function $l$ and $\mathbf{L}(z)= l(\Phi(z))(\max\{1,|\Phi_{z_1}'(z)|\},\ldots, \max\{1,|\Phi_{z_n}'(z)|\}),$ $z\in\mathbb{B}^n.$ Such a constructed function $\mathbf{L}$ allows us to consider a function $\Phi$ with a nonempty zero set for its gradient. The obtained results complement earlier published results with $\mathop{grad}\Phi(z)=(\frac{\partial \Phi(z)}{\partial z_1}, \ldots, \frac{\partial \Phi(z)}{\partial z_j},\ldots,\frac{\partial \Phi(z)}{\partial z_n})\ne \mathbf{0}.$ Also, we study a more general composition $H(\mathbf{w})=G(\mathbf{\Phi}(\mathbf{w}))$, where $G: \mathbb{C}^n\to \mathbb{C}$ is an entire function of the bounded $\mathbf{L}$-index in joint variables, $\mathbf{\Phi}: \mathbb{B}^m\to \mathbb{C}^n$ is a vector-valued analytic function, and $\mathbf{L}: \mathbb{C}^n\to\mathbb{R}^n_+$ is a continuous function. If the $\mathbf{L}$-index of the function $G$ equals zero, then we construct a function $\widetilde{\mathbf{L}}: \mathbb{B}^m\to\mathbb{R}^m_+$ such that the function $H$ has bounded $\widetilde{\mathbf{L}}$-index in the joint variables $w_1,$ $\ldots,$ $w_m$. These results are also new in one-dimensional case, i.e. for functions analytic in the unit disc.http://matstud.org.ua/ojs/index.php/matstud/article/view/578unit discunit ballanalytic functionentire functionseveral complex variablescompositionbounded index in joint variablessum of functionsgradient |
| spellingShingle | A. I. Bandura T. M. Salo O. B. Skaskiv Composition of entire function and analytic functions in the unit ball with a vanished gradient Математичні Студії unit disc unit ball analytic function entire function several complex variables composition bounded index in joint variables sum of functions gradient |
| title | Composition of entire function and analytic functions in the unit ball with a vanished gradient |
| title_full | Composition of entire function and analytic functions in the unit ball with a vanished gradient |
| title_fullStr | Composition of entire function and analytic functions in the unit ball with a vanished gradient |
| title_full_unstemmed | Composition of entire function and analytic functions in the unit ball with a vanished gradient |
| title_short | Composition of entire function and analytic functions in the unit ball with a vanished gradient |
| title_sort | composition of entire function and analytic functions in the unit ball with a vanished gradient |
| topic | unit disc unit ball analytic function entire function several complex variables composition bounded index in joint variables sum of functions gradient |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/578 |
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