解析一元多盘函数的有界L -方向索引

Q3 Mathematics
A. Bandura, T. Salo
{"title":"解析一元多盘函数的有界L -方向索引","authors":"A. Bandura, T. Salo","doi":"10.30970/ms.60.1.55-78","DOIUrl":null,"url":null,"abstract":"The concept of bounded $L$-index in a direction $\\mathbf{b}=(b_1,\\ldots,b_n)\\in\\mathbb{C}^n\\setminus\\{\\mathbf{0}\\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\\ldots,z_n)\\in\\mathbb{D}^n$ one has $L(z)>\\beta\\max_{1\\le j\\le n}\\frac{|b_j|}{1-|z_j|},$ $\\beta=\\mathrm{const}>1,$ $\\mathbb{D}^n$ is the unit polydisc, i.e. $\\mathbb{D}^n=\\{z\\in\\mathbb{C}^n: |z_j|\\le 1, j\\in\\{1,\\ldots,n\\}\\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\\{z+t\\mathbf{b}: |t|=r/L(z)\\}$ by their values at the center of the circle, where $t\\in\\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\\{z^0+t\\mathbf{b}: |t|\\le r/L(z^0)\\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Analytic in a unit polydisc functions of bounded $L$-index in direction\",\"authors\":\"A. Bandura, T. Salo\",\"doi\":\"10.30970/ms.60.1.55-78\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The concept of bounded $L$-index in a direction $\\\\mathbf{b}=(b_1,\\\\ldots,b_n)\\\\in\\\\mathbb{C}^n\\\\setminus\\\\{\\\\mathbf{0}\\\\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\\\\ldots,z_n)\\\\in\\\\mathbb{D}^n$ one has $L(z)>\\\\beta\\\\max_{1\\\\le j\\\\le n}\\\\frac{|b_j|}{1-|z_j|},$ $\\\\beta=\\\\mathrm{const}>1,$ $\\\\mathbb{D}^n$ is the unit polydisc, i.e. $\\\\mathbb{D}^n=\\\\{z\\\\in\\\\mathbb{C}^n: |z_j|\\\\le 1, j\\\\in\\\\{1,\\\\ldots,n\\\\}\\\\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\\\\{z+t\\\\mathbf{b}: |t|=r/L(z)\\\\}$ by their values at the center of the circle, where $t\\\\in\\\\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\\\\{z^0+t\\\\mathbf{b}: |t|\\\\le r/L(z^0)\\\\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-09-22\",\"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.60.1.55-78\",\"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.60.1.55-78","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

摘要

有界的概念 $L$-一个方向上的指数 $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ 是对单位多盘上的一类解析函数的推广,在哪里 $L$ 是否有连续函数对每一个都满足 $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ 一个是 $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ 是单位多盘,即。 $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ 对于该类的函数,我们得到了提供的有界性的充要条件 $L$-方向上的指数。他们描述了解析函数导数的最大模的局部性质 $F$ 在每个切片圆上 $\{z+t\mathbf{b}: |t|=r/L(z)\}$ 通过它们在圆心的值,其中 $t\in\mathbb{C}.$ 其他准则通过这些函数的最大模量描述了类似的最小模量的局部行为。我们证明了用函数描述对数导数在某例外集外的估计的对数准则的一个类比 $L$. 集合由所有切片盘的并集生成 $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$,其中 $z^0$ 是函数的零点吗 $F$. 模拟还表明了函数的零分布 $F$ 在所有切片圆盘上是均匀的。在一维情况下,该断言在微分方程的解析理论和无穷积(即Blaschke积、Naftalevich-Tsuji积)中有许多应用。对单位多面体上的解析函数,也推导出了海曼定理的类比。这表明在有界的定义中 $L$在索引方向上,可以去掉分母中的阶乘。这允许研究方向微分方程解析解的性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Analytic in a unit polydisc functions of bounded $L$-index in direction
The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信