The minimal growth of entire functions with given zeros along unbounded sets
Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition $$ \varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0 $$ is necessary and sufficient in order that for any complex sequence $(\zeta_n)$...
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| Format: | Article |
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Ivan Franko National University of Lviv
2020-12-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/160 |
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| author | I. V. Andrusyak P.V. Filevych |
| author_facet | I. V. Andrusyak P.V. Filevych |
| author_sort | I. V. Andrusyak |
| collection | DOAJ |
| description | Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition
$$
\varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0
$$
is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that
$$
\varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0.
$$
Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$. |
| format | Article |
| id | doaj-art-738e8677d735448094fe04e71dd267d0 |
| institution | DOAJ |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2020-12-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-738e8677d735448094fe04e71dd267d02025-08-20T03:17:40ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202020-12-0154214615310.30970/ms.54.2.146-153160The minimal growth of entire functions with given zeros along unbounded setsI. V. Andrusyak0P.V. Filevych1Lviv Polytechnic National University, Lviv, UkraineLviv Politechnic National University, LvivLet $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition $$ \varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0 $$ is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that $$ \varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0. $$ Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$.http://matstud.org.ua/ojs/index.php/matstud/article/view/160entire function; maximum modulus; zeros; counting function |
| spellingShingle | I. V. Andrusyak P.V. Filevych The minimal growth of entire functions with given zeros along unbounded sets Математичні Студії entire function; maximum modulus; zeros; counting function |
| title | The minimal growth of entire functions with given zeros along unbounded sets |
| title_full | The minimal growth of entire functions with given zeros along unbounded sets |
| title_fullStr | The minimal growth of entire functions with given zeros along unbounded sets |
| title_full_unstemmed | The minimal growth of entire functions with given zeros along unbounded sets |
| title_short | The minimal growth of entire functions with given zeros along unbounded sets |
| title_sort | minimal growth of entire functions with given zeros along unbounded sets |
| topic | entire function; maximum modulus; zeros; counting function |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/160 |
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