{"title":"关于奇数单值谐波映射","authors":"Kapil Jaglan, Anbareeswaran Sairam Kaliraj","doi":"10.1007/s13324-024-00964-5","DOIUrl":null,"url":null,"abstract":"<div><p>Odd univalent analytic functions played an instrumental role in the proof of the celebrated Bieberbach conjecture. In this article, we explore odd univalent harmonic mappings, focusing on coefficient estimates, growth and distortion theorems. Motivated by the unresolved harmonic analogue of the Bieberbach conjecture, we investigate specific subclasses of <span>\\({\\mathcal {S}}^0_H\\)</span>, the class of sense-preserving univalent harmonic functions. We provide sharp coefficient bounds for functions exhibiting convexity in one direction and extend our findings to a more generalized class including the major geometric subclasses of <span>\\({\\mathcal {S}}^0_H\\)</span>. Additionally, we analyze the inclusion of these functions in Hardy spaces and broaden the range of <i>p</i> for which they belong. In particular, the results of this article enhance understanding and highlight analogous growth patterns between odd univalent harmonic functions and the harmonic Bieberbach conjecture. We conclude the article with 2 conjectures and possible scope for further study as well.\n</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 5","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On odd univalent harmonic mappings\",\"authors\":\"Kapil Jaglan, Anbareeswaran Sairam Kaliraj\",\"doi\":\"10.1007/s13324-024-00964-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Odd univalent analytic functions played an instrumental role in the proof of the celebrated Bieberbach conjecture. In this article, we explore odd univalent harmonic mappings, focusing on coefficient estimates, growth and distortion theorems. Motivated by the unresolved harmonic analogue of the Bieberbach conjecture, we investigate specific subclasses of <span>\\\\({\\\\mathcal {S}}^0_H\\\\)</span>, the class of sense-preserving univalent harmonic functions. We provide sharp coefficient bounds for functions exhibiting convexity in one direction and extend our findings to a more generalized class including the major geometric subclasses of <span>\\\\({\\\\mathcal {S}}^0_H\\\\)</span>. Additionally, we analyze the inclusion of these functions in Hardy spaces and broaden the range of <i>p</i> for which they belong. In particular, the results of this article enhance understanding and highlight analogous growth patterns between odd univalent harmonic functions and the harmonic Bieberbach conjecture. We conclude the article with 2 conjectures and possible scope for further study as well.\\n</p></div>\",\"PeriodicalId\":48860,\"journal\":{\"name\":\"Analysis and Mathematical Physics\",\"volume\":\"14 5\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Mathematical Physics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s13324-024-00964-5\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s13324-024-00964-5","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
奇偶解析函数在证明著名的比伯巴赫猜想中发挥了重要作用。在这篇文章中,我们探讨了奇次不等式谐波映射,重点是系数估计、增长和畸变定理。在比伯巴赫猜想的未解谐波类比的激励下,我们研究了({\mathcal {S}}^0_H\)的特定子类,即保感单值谐函数类。我们为在一个方向上表现出凸性的函数提供了尖锐的系数边界,并将我们的发现扩展到一个更广义的类(包括 \({\mathcal {S}}^0_H\) 的主要几何子类)。此外,我们还分析了这些函数在哈代空间中的包含性,并拓宽了它们所属的 p 范围。特别是,本文的结果加深了对奇次单值谐函数与谐波比伯巴赫猜想之间类似增长模式的理解和强调。最后,我们还提出了两个猜想以及进一步研究的可能范围。
Odd univalent analytic functions played an instrumental role in the proof of the celebrated Bieberbach conjecture. In this article, we explore odd univalent harmonic mappings, focusing on coefficient estimates, growth and distortion theorems. Motivated by the unresolved harmonic analogue of the Bieberbach conjecture, we investigate specific subclasses of \({\mathcal {S}}^0_H\), the class of sense-preserving univalent harmonic functions. We provide sharp coefficient bounds for functions exhibiting convexity in one direction and extend our findings to a more generalized class including the major geometric subclasses of \({\mathcal {S}}^0_H\). Additionally, we analyze the inclusion of these functions in Hardy spaces and broaden the range of p for which they belong. In particular, the results of this article enhance understanding and highlight analogous growth patterns between odd univalent harmonic functions and the harmonic Bieberbach conjecture. We conclude the article with 2 conjectures and possible scope for further study as well.
期刊介绍:
Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.