{"title":"On Harnack inequality and harmonic Schwarz lemma","authors":"Rahim Kargar","doi":"10.4153/s0008439524000298","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529073251569-0683:S0008439524000298:S0008439524000298_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(s, C(s))$</span></span></img></span></span>-Harnack inequality in a domain <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529073251569-0683:S0008439524000298:S0008439524000298_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$G\\subset \\mathbb {R}^n$</span></span></img></span></span> for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529073251569-0683:S0008439524000298:S0008439524000298_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$s\\in (0,1)$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529073251569-0683:S0008439524000298:S0008439524000298_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$C(s)\\geq 1$</span></span></img></span></span> and present a series of inequalities related to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529073251569-0683:S0008439524000298:S0008439524000298_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$(s, C(s))$</span></span></img></span></span>-Harnack functions and the Harnack metric. We also investigate the behavior of the Harnack metric under <span>K</span>-quasiconformal and <span>K</span>-quasiregular mappings, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529073251569-0683:S0008439524000298:S0008439524000298_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$K\\geq 1$</span></span></img></span></span>. Finally, we provide a type of harmonic Schwarz lemma and improve the Schwarz–Pick estimate for a real-valued harmonic function.</p>","PeriodicalId":501184,"journal":{"name":"Canadian Mathematical Bulletin","volume":"29 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Mathematical Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4153/s0008439524000298","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the $(s, C(s))$-Harnack inequality in a domain $G\subset \mathbb {R}^n$ for $s\in (0,1)$ and $C(s)\geq 1$ and present a series of inequalities related to $(s, C(s))$-Harnack functions and the Harnack metric. We also investigate the behavior of the Harnack metric under K-quasiconformal and K-quasiregular mappings, where $K\geq 1$. Finally, we provide a type of harmonic Schwarz lemma and improve the Schwarz–Pick estimate for a real-valued harmonic function.