{"title":"具有准共形扩展的非零极谐波映射的面积定理","authors":"Bappaditya Bhowmik, Goutam Satpati","doi":"10.1090/proc/16850","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma Subscript upper H Superscript k Baseline left-parenthesis p right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msubsup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Sigma _H^k(p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the class of sense-preserving univalent harmonic mappings defined on the open unit disk <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper D\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"double-struck\">D</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the complex plane with a simple pole at <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"z equals p element-of left-parenthesis 0 comma 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">z=p \\in (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that have <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-quasiconformal extensions (<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 less-than-or-equal-to k greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">0\\leq k>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) onto the extended complex plane. In this article, we obtain an area theorem for this class of functions.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"53 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An area theorem for harmonic mappings with nonzero pole having quasiconformal extensions\",\"authors\":\"Bappaditya Bhowmik, Goutam Satpati\",\"doi\":\"10.1090/proc/16850\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Sigma Subscript upper H Superscript k Baseline left-parenthesis p right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant=\\\"normal\\\">Σ</mml:mi> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msubsup> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Sigma _H^k(p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the class of sense-preserving univalent harmonic mappings defined on the open unit disk <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper D\\\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">D</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the complex plane with a simple pole at <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"z equals p element-of left-parenthesis 0 comma 1 right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">z=p \\\\in (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that have <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"k\\\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-quasiconformal extensions (<inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"0 less-than-or-equal-to k greater-than 1\\\"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">0\\\\leq k>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) onto the extended complex plane. In this article, we obtain an area theorem for this class of functions.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"53 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-03-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16850\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16850","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设 Σ H k ( p ) \Sigma _H^k(p)是一类定义在复平面的开放单位盘 D \mathbb {D} 上、在 z = p∈ ( 0 , 1 ) z=p \(0,1)处有一个简单极点、在扩展复平面上有 k k -等方扩展（0 ≤ k > 1 0\leq k>1 ）的保感单等调和映射。在本文中，我们得到了这一类函数的面积定理。
An area theorem for harmonic mappings with nonzero pole having quasiconformal extensions
Let ΣHk(p)\Sigma _H^k(p) be the class of sense-preserving univalent harmonic mappings defined on the open unit disk D\mathbb {D} of the complex plane with a simple pole at z=p∈(0,1)z=p \in (0,1) that have kk-quasiconformal extensions (0≤k>10\leq k>1) onto the extended complex plane. In this article, we obtain an area theorem for this class of functions.
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