{"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}
引用次数: 0
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$$m_2(r,f)=\left(\frac{1}{2\pi}\int_0^{2\pi}|\ln|f(re^{i\theta})||^2d\theta\right)^{1/2}.$$