{"title":"Entire Bivariate Functions of Exponential Type II","authors":"Andriy Ivanovych Bandura, F. Nuray","doi":"10.30970/ms.59.2.156-167","DOIUrl":null,"url":null,"abstract":"Let $f(z_{1},z_{2})$ be a bivariate entire function and $C$ be a positive constant. If $f(z_{1},z_{2})$ satisfies the following inequality for non-negative integer $M$, for all non-negative integers $k,$ $l$ such that $k+l\\in\\{0, 1, 2, \\ldots, M\\}$, for some integer $p\\ge 1$ and for all $(z_{1},z_{2})=(r_{1}e^{\\mathbf{i}\\theta_{1}},r_{2}e^{\\mathbf{i}\\theta_{2}})$ with $r_1$ and $r_2$ sufficiently large:\\begin{gather*}\\sum_{i+j=0}^{M}\\frac{\\left(\\int_{0}^{2\\pi}\\int_{0}^{2\\pi}|f^{(i+k,j+l)}(r_{1}e^{\\mathbf{i}\\theta_{1}},r_{2}e^{\\mathbf{i}\\theta_{2}})|^{p}d\\theta_{1}\\theta_{2}\\right)^{\\frac{1}{p}}}{i!j!}\\ge \\\\\\ge \\sum_{i+j=M+1}^{\\infty}\\frac{\\left(\\int_{0}^{2\\pi}\\int_{0}^{2\\pi}|f^{(i+k,j+l)}(r_{1}e^{\\mathbf{i}\\theta_{1}},r_{2}e^{\\mathbf{i}\\theta_{2}})|^{p}d\\theta_{1}\\theta_{2}\\right)^{\\frac{1}{p}}}{i!j!},\\end{gather*}then $f(z_{1},z_{2})$ is of exponential type not exceeding\\[2+2\\log\\Big(1+\\frac{1}{C}\\Big)+\\log[(2M)!/M!].\\]If this condition is replaced by related conditions, then also $f$ is of exponential type.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.59.2.156-167","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: 2
Abstract
Let $f(z_{1},z_{2})$ be a bivariate entire function and $C$ be a positive constant. If $f(z_{1},z_{2})$ satisfies the following inequality for non-negative integer $M$, for all non-negative integers $k,$ $l$ such that $k+l\in\{0, 1, 2, \ldots, M\}$, for some integer $p\ge 1$ and for all $(z_{1},z_{2})=(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})$ with $r_1$ and $r_2$ sufficiently large:\begin{gather*}\sum_{i+j=0}^{M}\frac{\left(\int_{0}^{2\pi}\int_{0}^{2\pi}|f^{(i+k,j+l)}(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})|^{p}d\theta_{1}\theta_{2}\right)^{\frac{1}{p}}}{i!j!}\ge \\\ge \sum_{i+j=M+1}^{\infty}\frac{\left(\int_{0}^{2\pi}\int_{0}^{2\pi}|f^{(i+k,j+l)}(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})|^{p}d\theta_{1}\theta_{2}\right)^{\frac{1}{p}}}{i!j!},\end{gather*}then $f(z_{1},z_{2})$ is of exponential type not exceeding\[2+2\log\Big(1+\frac{1}{C}\Big)+\log[(2M)!/M!].\]If this condition is replaced by related conditions, then also $f$ is of exponential type.