On the interpolation in some classes of holomorphic in the unit disk functions
There is considered an interpolation problem $f(\lambda_n )=b_n$ in the class of holomorphic in the unit disk $U(0;1)=\{z\in\mathbb{C}\colon |z|<1\}$ functions of finite $\eta$-type, i.e such that $\displaystyle (\exists A>0)(\forall z\in U(0;1))\colon \quad |f(z)|\leq\exp\Big(A\eta\Big(\fr...
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Ivan Franko National University of Lviv
2024-09-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/483 |
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| author | I.B. Sheparovych |
| author_facet | I.B. Sheparovych |
| author_sort | I.B. Sheparovych |
| collection | DOAJ |
| description | There is considered an interpolation problem $f(\lambda_n )=b_n$ in the class of holomorphic in the unit disk $U(0;1)=\{z\in\mathbb{C}\colon |z|<1\}$
functions of finite $\eta$-type, i.e such that
$\displaystyle (\exists A>0)(\forall z\in U(0;1))\colon \quad |f(z)|\leq\exp\Big(A\eta\Big(\frac A{1-|z|}\Big)\Big),$
where $\eta\colon [1;+\infty)\to [0;+\infty)$ is an increasing convex function with respect to $\ln{t}$ and $\ln{t}=o\left(\eta ( t)\right)$ $(t\to+\infty)$.
There were received sufficient conditions of the interpolation problem solvability in terms of the counting functions
$\displaystyle N(r)=\int\nolimits_{0}^{r}\frac{\left(n(t)-1\right)^+}{t}dt$ and $\displaystyle N_{\lambda_n} (r)=\int\nolimits_{0}^{r}{\frac{{{(n}_{\lambda_n}\left(t\right)-1)}^+}{t}dt}$.
Earlier, in 2004, necessary conditions were obtained (Ukr. Math. J., {\bf 56} (2004), \No 3) in these terms.
For the moderate growth of $f$ (when the majorant $\eta=\psi$ satisfies the condition $\psi\left(2x\right)=O\left(\psi\left(x\right)\right),\ x\rightarrow+\infty$) that problem was solved in J. Math. Anal. Appl., {\bf 414} (2014), \No 1.
In this paper, we remove any restrictions on the growth of $\eta$ and construct an interpolation function $f$ such that
$\displaystyle (\exists A'>0)(\forall z\in U(0;1))\colon \quad |{f}(z)|\leq\exp\Big(\frac{A'}{(1-|z|)^{3/2}}\eta\Big(\frac{A'}{1-|z|}\Big)\Big)$. |
| format | Article |
| id | doaj-art-26b73063533e4ff489ad5cb67c5342a4 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2024-09-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-26b73063533e4ff489ad5cb67c5342a42025-08-20T03:33:17ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-09-01621313810.30970/ms.62.1.31-38483On the interpolation in some classes of holomorphic in the unit disk functionsI.B. Sheparovych0Drohobych Ivan Franko State Pedagogical University Drohobych, UkraineThere is considered an interpolation problem $f(\lambda_n )=b_n$ in the class of holomorphic in the unit disk $U(0;1)=\{z\in\mathbb{C}\colon |z|<1\}$ functions of finite $\eta$-type, i.e such that $\displaystyle (\exists A>0)(\forall z\in U(0;1))\colon \quad |f(z)|\leq\exp\Big(A\eta\Big(\frac A{1-|z|}\Big)\Big),$ where $\eta\colon [1;+\infty)\to [0;+\infty)$ is an increasing convex function with respect to $\ln{t}$ and $\ln{t}=o\left(\eta ( t)\right)$ $(t\to+\infty)$. There were received sufficient conditions of the interpolation problem solvability in terms of the counting functions $\displaystyle N(r)=\int\nolimits_{0}^{r}\frac{\left(n(t)-1\right)^+}{t}dt$ and $\displaystyle N_{\lambda_n} (r)=\int\nolimits_{0}^{r}{\frac{{{(n}_{\lambda_n}\left(t\right)-1)}^+}{t}dt}$. Earlier, in 2004, necessary conditions were obtained (Ukr. Math. J., {\bf 56} (2004), \No 3) in these terms. For the moderate growth of $f$ (when the majorant $\eta=\psi$ satisfies the condition $\psi\left(2x\right)=O\left(\psi\left(x\right)\right),\ x\rightarrow+\infty$) that problem was solved in J. Math. Anal. Appl., {\bf 414} (2014), \No 1. In this paper, we remove any restrictions on the growth of $\eta$ and construct an interpolation function $f$ such that $\displaystyle (\exists A'>0)(\forall z\in U(0;1))\colon \quad |{f}(z)|\leq\exp\Big(\frac{A'}{(1-|z|)^{3/2}}\eta\Big(\frac{A'}{1-|z|}\Big)\Big)$.http://matstud.org.ua/ojs/index.php/matstud/article/view/483interpolation problemholomorphic functionunit diskzeros sequence |
| spellingShingle | I.B. Sheparovych On the interpolation in some classes of holomorphic in the unit disk functions Математичні Студії interpolation problem holomorphic function unit disk zeros sequence |
| title | On the interpolation in some classes of holomorphic in the unit disk functions |
| title_full | On the interpolation in some classes of holomorphic in the unit disk functions |
| title_fullStr | On the interpolation in some classes of holomorphic in the unit disk functions |
| title_full_unstemmed | On the interpolation in some classes of holomorphic in the unit disk functions |
| title_short | On the interpolation in some classes of holomorphic in the unit disk functions |
| title_sort | on the interpolation in some classes of holomorphic in the unit disk functions |
| topic | interpolation problem holomorphic function unit disk zeros sequence |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/483 |
| work_keys_str_mv | AT ibsheparovych ontheinterpolationinsomeclassesofholomorphicintheunitdiskfunctions |