Analytic in the unit polydisc functions of bounded L-index in direction
The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\b...
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Ivan Franko National University of Lviv
2023-09-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/450 |
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| author | A. Bandura T. Salo |
| author_facet | A. Bandura T. Salo |
| author_sort | A. Bandura |
| collection | DOAJ |
| description | The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations. |
| format | Article |
| id | doaj-art-fb9692b34a3f473c88712e9714578da3 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2023-09-01 |
| publisher | Ivan Franko National University of Lviv |
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| series | Математичні Студії |
| spelling | doaj-art-fb9692b34a3f473c88712e9714578da32025-08-20T03:28:41ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202023-09-01601557810.30970/ms.60.1.55-78450Analytic in the unit polydisc functions of bounded L-index in directionA. Bandura0T. Salo1Ivano-Frankivsk National Tecnical University of OIl and GasLviv Politechnic National UniversityThe concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.http://matstud.org.ua/ojs/index.php/matstud/article/view/450analytic function; unit polydisc; bounded $l$-index in direction; directional derivative; maximum modulus; minimum modulus; logarithmic derivative; zero distribution; local behavior |
| spellingShingle | A. Bandura T. Salo Analytic in the unit polydisc functions of bounded L-index in direction Математичні Студії analytic function; unit polydisc; bounded $l$-index in direction; directional derivative; maximum modulus; minimum modulus; logarithmic derivative; zero distribution; local behavior |
| title | Analytic in the unit polydisc functions of bounded L-index in direction |
| title_full | Analytic in the unit polydisc functions of bounded L-index in direction |
| title_fullStr | Analytic in the unit polydisc functions of bounded L-index in direction |
| title_full_unstemmed | Analytic in the unit polydisc functions of bounded L-index in direction |
| title_short | Analytic in the unit polydisc functions of bounded L-index in direction |
| title_sort | analytic in the unit polydisc functions of bounded l index in direction |
| topic | analytic function; unit polydisc; bounded $l$-index in direction; directional derivative; maximum modulus; minimum modulus; logarithmic derivative; zero distribution; local behavior |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/450 |
| work_keys_str_mv | AT abandura analyticintheunitpolydiscfunctionsofboundedlindexindirection AT tsalo analyticintheunitpolydiscfunctionsofboundedlindexindirection |