Asymptotic estimates for entire functions of minimal growth with given zeros
Let $\zeta=(\zeta_n)$ be an arbitrary complex sequence such that $0<|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, let $n_\zeta(r)$ and $N_\zeta(r)$ be the counting function and the integrated counting function of this sequence, respectively. By $\mathcal{E}_\zeta$ we deno...
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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/552 |
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| author | P. V. Filevych |
| author_facet | P. V. Filevych |
| author_sort | P. V. Filevych |
| collection | DOAJ |
| description | Let $\zeta=(\zeta_n)$ be an arbitrary complex sequence such that $0<|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, let $n_\zeta(r)$ and $N_\zeta(r)$ be the counting function and the integrated counting function of this sequence, respectively. By $\mathcal{E}_\zeta$ we denote the class of all entire functions whose zeros are precisely the $\zeta_n$, where a complex number that occurs $m$ times in the sequence $\zeta$ corresponds to a zero of multiplicity $m$. Suppose that $\Phi$ is a convex function on $\mathbb{R}$ such that $\Phi(\sigma)/\sigma\to+\infty$ as $\sigma\to+\infty$. It is proved that there exists an entire function $f\in\mathcal{E}_\zeta$ such that
$$
\varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln n_\zeta( r)}{\Phi(\ln r)},
$$
where $M_f(r)$ denotes the maximum modulus of the function $f$, and it is shown that the above inequality implies the inequality
$$
\varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln N_\zeta( r)}{\Phi(\ln r)}+\varlimsup_{\sigma\to+\infty}\frac{\ln\Phi'_+(\sigma)}{\Phi(\sigma)}.
$$
The formulated result is a consequence of the following more general statement: if the right-hand derivative $\Phi'_+$ of the function $\Phi$ assumes only integer values and $\sum_{n=1}^\infty e^{-\Phi(\ln|\zeta_n|)}<+\infty$, then there exists an entire function $f\in\mathcal{E}_\zeta$ such that $\ln M_f(r)=o(e^{\Phi(\ln r)})$ as $r \to+\infty$. |
| format | Article |
| id | doaj-art-1b69be99a6794ea4820c42bc26e05702 |
| 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-1b69be99a6794ea4820c42bc26e057022025-08-20T03:28:21ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-09-01621545910.30970/ms.62.1.54-59552Asymptotic estimates for entire functions of minimal growth with given zerosP. V. Filevych0Lviv Polytechnic National University Lviv, UkraineLet $\zeta=(\zeta_n)$ be an arbitrary complex sequence such that $0<|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, let $n_\zeta(r)$ and $N_\zeta(r)$ be the counting function and the integrated counting function of this sequence, respectively. By $\mathcal{E}_\zeta$ we denote the class of all entire functions whose zeros are precisely the $\zeta_n$, where a complex number that occurs $m$ times in the sequence $\zeta$ corresponds to a zero of multiplicity $m$. Suppose that $\Phi$ is a convex function on $\mathbb{R}$ such that $\Phi(\sigma)/\sigma\to+\infty$ as $\sigma\to+\infty$. It is proved that there exists an entire function $f\in\mathcal{E}_\zeta$ such that $$ \varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln n_\zeta( r)}{\Phi(\ln r)}, $$ where $M_f(r)$ denotes the maximum modulus of the function $f$, and it is shown that the above inequality implies the inequality $$ \varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln N_\zeta( r)}{\Phi(\ln r)}+\varlimsup_{\sigma\to+\infty}\frac{\ln\Phi'_+(\sigma)}{\Phi(\sigma)}. $$ The formulated result is a consequence of the following more general statement: if the right-hand derivative $\Phi'_+$ of the function $\Phi$ assumes only integer values and $\sum_{n=1}^\infty e^{-\Phi(\ln|\zeta_n|)}<+\infty$, then there exists an entire function $f\in\mathcal{E}_\zeta$ such that $\ln M_f(r)=o(e^{\Phi(\ln r)})$ as $r \to+\infty$.http://matstud.org.ua/ojs/index.php/matstud/article/view/552entire functionmaximum modulusorderzero;counting functionintegrated counting function |
| spellingShingle | P. V. Filevych Asymptotic estimates for entire functions of minimal growth with given zeros Математичні Студії entire function maximum modulus order zero; counting function integrated counting function |
| title | Asymptotic estimates for entire functions of minimal growth with given zeros |
| title_full | Asymptotic estimates for entire functions of minimal growth with given zeros |
| title_fullStr | Asymptotic estimates for entire functions of minimal growth with given zeros |
| title_full_unstemmed | Asymptotic estimates for entire functions of minimal growth with given zeros |
| title_short | Asymptotic estimates for entire functions of minimal growth with given zeros |
| title_sort | asymptotic estimates for entire functions of minimal growth with given zeros |
| topic | entire function maximum modulus order zero; counting function integrated counting function |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/552 |
| work_keys_str_mv | AT pvfilevych asymptoticestimatesforentirefunctionsofminimalgrowthwithgivenzeros |