Dong Guo, Huo Tang, Jun Zhang, Qingbing Xu, Zongtao Li
{"title":"近星函数子类对数系数的汉克尔决定因素","authors":"Dong Guo, Huo Tang, Jun Zhang, Qingbing Xu, Zongtao Li","doi":"10.1155/2024/1315252","DOIUrl":null,"url":null,"abstract":"Suppose that <svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 36.9885 11.5564\" width=\"36.9885pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,9.659,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,21.515,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,26.013,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,32.253,0)\"></path></g></svg> is a class of close-to-star functions. In this paper, we investigated the estimate of Zalcman functional on the logarithmic coefficients and the third Hankel determinant for the class <svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 36.9885 11.5564\" width=\"36.9885pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-20\"></use></g><g transform=\"matrix(.013,0,0,-0.013,9.659,0)\"><use xlink:href=\"#g198-21\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.515,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,26.013,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,32.253,0)\"><use xlink:href=\"#g113-42\"></use></g></svg> with the determinant entry of logarithmic coefficients. Also, we obtained the sharp bounds of Zalcman functional <svg height=\"13.639pt\" style=\"vertical-align:-4.35067pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 34.116 13.639\" width=\"34.116pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,5.031,3.132)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,9.463,3.132)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,11.619,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,16.59,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.088,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,29.441,0)\"><use xlink:href=\"#g113-42\"></use></g></svg> and <svg height=\"13.639pt\" style=\"vertical-align:-4.35067pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 34.116 13.639\" width=\"34.116pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-75\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,5.031,3.132)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,9.463,3.132)\"><use xlink:href=\"#g50-45\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,11.62,3.132)\"><use xlink:href=\"#g50-52\"></use></g><g transform=\"matrix(.013,0,0,-0.013,16.59,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.088,0)\"><use xlink:href=\"#g113-103\"></use></g><g transform=\"matrix(.013,0,0,-0.013,29.441,0)\"><use xlink:href=\"#g113-42\"></use></g></svg> for the class <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 36.9885 11.5564\" width=\"36.9885pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-20\"></use></g><g transform=\"matrix(.013,0,0,-0.013,9.659,0)\"><use xlink:href=\"#g198-21\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.515,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,26.013,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,32.253,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>.</span>","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"14 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hankel Determinants for the Logarithmic Coefficients of a Subclass of Close-to-Star Functions\",\"authors\":\"Dong Guo, Huo Tang, Jun Zhang, Qingbing Xu, Zongtao Li\",\"doi\":\"10.1155/2024/1315252\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Suppose that <svg height=\\\"11.5564pt\\\" style=\\\"vertical-align:-2.26807pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 36.9885 11.5564\\\" width=\\\"36.9885pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,9.659,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,21.515,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,26.013,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,32.253,0)\\\"></path></g></svg> is a class of close-to-star functions. In this paper, we investigated the estimate of Zalcman functional on the logarithmic coefficients and the third Hankel determinant for the class <svg height=\\\"11.5564pt\\\" style=\\\"vertical-align:-2.26807pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 36.9885 11.5564\\\" width=\\\"36.9885pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g198-20\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,9.659,0)\\\"><use xlink:href=\\\"#g198-21\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,21.515,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,26.013,0)\\\"><use xlink:href=\\\"#g113-50\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,32.253,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g></svg> with the determinant entry of logarithmic coefficients. Also, we obtained the sharp bounds of Zalcman functional <svg height=\\\"13.639pt\\\" style=\\\"vertical-align:-4.35067pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 34.116 13.639\\\" width=\\\"34.116pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,5.031,3.132)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,9.463,3.132)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,11.619,3.132)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,16.59,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,21.088,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,29.441,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g></svg> and <svg height=\\\"13.639pt\\\" style=\\\"vertical-align:-4.35067pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 34.116 13.639\\\" width=\\\"34.116pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-75\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,5.031,3.132)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,9.463,3.132)\\\"><use xlink:href=\\\"#g50-45\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,11.62,3.132)\\\"><use xlink:href=\\\"#g50-52\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,16.59,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,21.088,0)\\\"><use xlink:href=\\\"#g113-103\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,29.441,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g></svg> for the class <span><svg height=\\\"11.5564pt\\\" style=\\\"vertical-align:-2.26807pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 36.9885 11.5564\\\" width=\\\"36.9885pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g198-20\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,9.659,0)\\\"><use xlink:href=\\\"#g198-21\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,21.515,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,26.013,0)\\\"><use xlink:href=\\\"#g113-50\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,32.253,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g></svg>.</span>\",\"PeriodicalId\":54214,\"journal\":{\"name\":\"Journal of Mathematics\",\"volume\":\"14 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-05-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1155/2024/1315252\",\"RegionNum\":4,\"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":"Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/1315252","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Hankel Determinants for the Logarithmic Coefficients of a Subclass of Close-to-Star Functions
Suppose that is a class of close-to-star functions. In this paper, we investigated the estimate of Zalcman functional on the logarithmic coefficients and the third Hankel determinant for the class with the determinant entry of logarithmic coefficients. Also, we obtained the sharp bounds of Zalcman functional and for the class .
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Journal of Mathematics is a broad scope journal that publishes original research articles as well as review articles on all aspects of both pure and applied mathematics. As well as original research, Journal of Mathematics also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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