{"title":"φ正态谐波映射的拉潘五点定理","authors":"Nisha Bohra, Gopal Datt, Ritesh Pal","doi":"arxiv-2408.05809","DOIUrl":null,"url":null,"abstract":"A harmonic mapping $f=h+\\overline{g}$ in $\\mathbb{D}$ is $\\varphi$-normal if\n$f^{\\#}(z)=\\mathcal{O}(|\\varphi(z)|), \\text{ as } |z|\\to 1^-,$ where\n$f^{\\#}(z)={(|h'(z)|+|g'(z)|)}/{(1+|f(z)|^2)}.$ In this paper, we establish\nseveral sufficient conditions for harmonic mappings to be $\\varphi$-normal. We\nalso extend the five-point theorem of Lappan for $\\varphi$-normal harmonic\nmappings.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"7 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Lappan's five-point theorem for φ-Normal Harmonic Mappings\",\"authors\":\"Nisha Bohra, Gopal Datt, Ritesh Pal\",\"doi\":\"arxiv-2408.05809\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A harmonic mapping $f=h+\\\\overline{g}$ in $\\\\mathbb{D}$ is $\\\\varphi$-normal if\\n$f^{\\\\#}(z)=\\\\mathcal{O}(|\\\\varphi(z)|), \\\\text{ as } |z|\\\\to 1^-,$ where\\n$f^{\\\\#}(z)={(|h'(z)|+|g'(z)|)}/{(1+|f(z)|^2)}.$ In this paper, we establish\\nseveral sufficient conditions for harmonic mappings to be $\\\\varphi$-normal. We\\nalso extend the five-point theorem of Lappan for $\\\\varphi$-normal harmonic\\nmappings.\",\"PeriodicalId\":501142,\"journal\":{\"name\":\"arXiv - MATH - Complex Variables\",\"volume\":\"7 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Complex Variables\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.05809\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.05809","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Lappan's five-point theorem for φ-Normal Harmonic Mappings
A harmonic mapping $f=h+\overline{g}$ in $\mathbb{D}$ is $\varphi$-normal if
$f^{\#}(z)=\mathcal{O}(|\varphi(z)|), \text{ as } |z|\to 1^-,$ where
$f^{\#}(z)={(|h'(z)|+|g'(z)|)}/{(1+|f(z)|^2)}.$ In this paper, we establish
several sufficient conditions for harmonic mappings to be $\varphi$-normal. We
also extend the five-point theorem of Lappan for $\varphi$-normal harmonic
mappings.
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