{"title":"例外集外具有指定零的整函数的最小增长","authors":"I. Andrusyak, P. Filevych, O. Oryshchyn","doi":"10.30970/ms.58.1.51-57","DOIUrl":null,"url":null,"abstract":"Let $h$ be a positive continuous increasing to $+\\infty$ function on $\\mathbb{R}$. It is proved that for an arbitrary complex sequence $(\\zeta_n)$ such that $0<|\\zeta_1|\\le|\\zeta_2|\\le\\dots$ and $\\zeta_n\\to\\infty$ as $n\\to\\infty$, there exists an entire function $f$ whose zeros are the $\\zeta_n$, with multiplicities taken into account, for which$$\\ln m_2(r,f)=o(N(r)),\\quad r\\notin E,\\ r\\to+\\infty.$$with a set $E$ satisfying $\\int_{E\\cap(1,+\\infty)}h(r)dr<+\\infty$, if and only if $\\ln h(r)=O(\\ln r)$ as $r\\to+\\infty$.Here $N(r)$ is the integrated counting function of the sequence $(\\zeta_n)$ and$$m_2(r,f)=\\left(\\frac{1}{2\\pi}\\int_0^{2\\pi}|\\ln|f(re^{i\\theta})||^2d\\theta\\right)^{1/2}.$$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Minimal growth of entire functions with prescribed zeros outside exceptional sets\",\"authors\":\"I. Andrusyak, P. Filevych, O. Oryshchyn\",\"doi\":\"10.30970/ms.58.1.51-57\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $h$ be a positive continuous increasing to $+\\\\infty$ function on $\\\\mathbb{R}$. It is proved that for an arbitrary complex sequence $(\\\\zeta_n)$ such that $0<|\\\\zeta_1|\\\\le|\\\\zeta_2|\\\\le\\\\dots$ and $\\\\zeta_n\\\\to\\\\infty$ as $n\\\\to\\\\infty$, there exists an entire function $f$ whose zeros are the $\\\\zeta_n$, with multiplicities taken into account, for which$$\\\\ln m_2(r,f)=o(N(r)),\\\\quad r\\\\notin E,\\\\ r\\\\to+\\\\infty.$$with a set $E$ satisfying $\\\\int_{E\\\\cap(1,+\\\\infty)}h(r)dr<+\\\\infty$, if and only if $\\\\ln h(r)=O(\\\\ln r)$ as $r\\\\to+\\\\infty$.Here $N(r)$ is the integrated counting function of the sequence $(\\\\zeta_n)$ and$$m_2(r,f)=\\\\left(\\\\frac{1}{2\\\\pi}\\\\int_0^{2\\\\pi}|\\\\ln|f(re^{i\\\\theta})||^2d\\\\theta\\\\right)^{1/2}.$$\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-10-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Matematychni Studii\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30970/ms.58.1.51-57\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.58.1.51-57","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
Minimal growth of entire functions with prescribed zeros outside exceptional sets
Let $h$ be a positive continuous increasing to $+\infty$ function on $\mathbb{R}$. It is proved that for an arbitrary complex sequence $(\zeta_n)$ such that $0<|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, there exists an entire function $f$ whose zeros are the $\zeta_n$, with multiplicities taken into account, for which$$\ln m_2(r,f)=o(N(r)),\quad r\notin E,\ r\to+\infty.$$with a set $E$ satisfying $\int_{E\cap(1,+\infty)}h(r)dr<+\infty$, if and only if $\ln h(r)=O(\ln r)$ as $r\to+\infty$.Here $N(r)$ is the integrated counting function of the sequence $(\zeta_n)$ and$$m_2(r,f)=\left(\frac{1}{2\pi}\int_0^{2\pi}|\ln|f(re^{i\theta})||^2d\theta\right)^{1/2}.$$