{"title":"On the Hyperbolic Metric of Certain Domains","authors":"Aimo Hinkkanen, Matti Vuorinen","doi":"10.1007/s40315-023-00518-z","DOIUrl":null,"url":null,"abstract":"<p>We prove that if <i>E</i> is a compact subset of the unit disk <span>\\({{\\mathbb {D}}}\\)</span> in the complex plane, if <i>E</i> contains a sequence of distinct points <span>\\(a_n\\not = 0\\)</span> for <span>\\(n\\ge 1\\)</span> such that <span>\\(\\lim _{n\\rightarrow \\infty } a_n=0\\)</span> and for all <i>n</i> we have <span>\\( |a_{n+1}| \\ge |a_n|/2 \\)</span>, and if <span>\\(G={{\\mathbb {D}}} {\\setminus } E\\)</span> is connected and <span>\\(0\\in \\partial G\\)</span>, then there is a constant <span>\\(c>0\\)</span> such that for all <span>\\(z\\in G\\)</span> we have <span>\\( \\lambda _{G } (z) \\ge c/|z| \\)</span> where <span>\\(\\lambda _{G } (z)\\)</span> is the density of the hyperbolic metric in <i>G</i>.</p>","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"108 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods and Function Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-023-00518-z","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that if E is a compact subset of the unit disk \({{\mathbb {D}}}\) in the complex plane, if E contains a sequence of distinct points \(a_n\not = 0\) for \(n\ge 1\) such that \(\lim _{n\rightarrow \infty } a_n=0\) and for all n we have \( |a_{n+1}| \ge |a_n|/2 \), and if \(G={{\mathbb {D}}} {\setminus } E\) is connected and \(0\in \partial G\), then there is a constant \(c>0\) such that for all \(z\in G\) we have \( \lambda _{G } (z) \ge c/|z| \) where \(\lambda _{G } (z)\) is the density of the hyperbolic metric in G.
我们证明,如果 E 是复平面上单位盘 \({{\mathbb {D}}}\) 的一个紧凑子集,如果 E 包含一系列不同点 \(a_n\not = 0\) for \(n\ge 1\) such that \(\lim _{n\rightarrow \infty } a_n=0\) and for all n we have \( |a_{n+1}| \ge |a_n|/2 \)、如果(G={{\mathbb {D}}} {\setminus } E\ )是连通的,并且(0\in \partial G\ ),那么有一个常数(c>;0),这样对于所有的(z在G中),我们都有\( \lambda _{G } (z)\ge c/(z) \ge c/|z| \) where \(\lambda _{G } (z)\) is the means of the G.(z)\) 是 G 中双曲度量的密度。
期刊介绍:
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.