{"title":"将单位圆盘映射到伯努利矩阵内部的解析函数","authors":"Shalu Yadav, Vaithiyanathan Ravichandran","doi":"10.3934/mfc.2022036","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>The function <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\varphi_L $\\end{document}</tex-math></inline-formula> defined by <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\varphi_L(z) = \\sqrt{1+z} $\\end{document}</tex-math></inline-formula> maps the unit disk <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\mathbb{D} $\\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id=\"M8\">\\begin{document}$ \\Omega = \\{w\\in\\mathbb{C}: |w^2-1|<1\\} $\\end{document}</tex-math></inline-formula>, the region in the right half-plane bounded by the lemniscate of Bernoulli <inline-formula><tex-math id=\"M9\">\\begin{document}$ |w^2-1| = 1 $\\end{document}</tex-math></inline-formula>. This paper deals with starlike functions defined on <inline-formula><tex-math id=\"M10\">\\begin{document}$ \\mathbb{D} $\\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M11\">\\begin{document}$ zf'(z)/f(z)\\in \\Omega $\\end{document}</tex-math></inline-formula> or equivalently <inline-formula><tex-math id=\"M12\">\\begin{document}$ zf'(z)/f(z) $\\end{document}</tex-math></inline-formula> is subordinated to <inline-formula><tex-math id=\"M13\">\\begin{document}$ \\varphi_L(z) $\\end{document}</tex-math></inline-formula> and these functions are related to the analytic function <inline-formula><tex-math id=\"M14\">\\begin{document}$ p:\\mathbb{D}\\to \\mathbb{C} $\\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M15\">\\begin{document}$ p(z)\\in \\Omega $\\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id=\"M16\">\\begin{document}$ z\\in \\mathbb{D} $\\end{document}</tex-math></inline-formula> by <inline-formula><tex-math id=\"M17\">\\begin{document}$ p(z) = zf'(z)/f(z) $\\end{document}</tex-math></inline-formula>. Using the admissibility criteria of the first and second order differential subordination, we investigate several subordination results for functions <inline-formula><tex-math id=\"M18\">\\begin{document}$ p $\\end{document}</tex-math></inline-formula> to satisfy <inline-formula><tex-math id=\"M19\">\\begin{document}$ p(z)\\in \\Omega $\\end{document}</tex-math></inline-formula>. As applications, we give several sufficient conditions for functions <inline-formula><tex-math id=\"M20\">\\begin{document}$ f $\\end{document}</tex-math></inline-formula> to satisfy <inline-formula><tex-math id=\"M21\">\\begin{document}$ zf'(z)/f(z)\\in \\Omega $\\end{document}</tex-math></inline-formula>.</p>","PeriodicalId":93334,"journal":{"name":"Mathematical foundations of computing","volume":"7 1","pages":"591-600"},"PeriodicalIF":1.3000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Analytic function that map the unit disk into the inside of the lemniscate of Bernoulli\",\"authors\":\"Shalu Yadav, Vaithiyanathan Ravichandran\",\"doi\":\"10.3934/mfc.2022036\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>The function <inline-formula><tex-math id=\\\"M5\\\">\\\\begin{document}$ \\\\varphi_L $\\\\end{document}</tex-math></inline-formula> defined by <inline-formula><tex-math id=\\\"M6\\\">\\\\begin{document}$ \\\\varphi_L(z) = \\\\sqrt{1+z} $\\\\end{document}</tex-math></inline-formula> maps the unit disk <inline-formula><tex-math id=\\\"M7\\\">\\\\begin{document}$ \\\\mathbb{D} $\\\\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id=\\\"M8\\\">\\\\begin{document}$ \\\\Omega = \\\\{w\\\\in\\\\mathbb{C}: |w^2-1|<1\\\\} $\\\\end{document}</tex-math></inline-formula>, the region in the right half-plane bounded by the lemniscate of Bernoulli <inline-formula><tex-math id=\\\"M9\\\">\\\\begin{document}$ |w^2-1| = 1 $\\\\end{document}</tex-math></inline-formula>. This paper deals with starlike functions defined on <inline-formula><tex-math id=\\\"M10\\\">\\\\begin{document}$ \\\\mathbb{D} $\\\\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\\\"M11\\\">\\\\begin{document}$ zf'(z)/f(z)\\\\in \\\\Omega $\\\\end{document}</tex-math></inline-formula> or equivalently <inline-formula><tex-math id=\\\"M12\\\">\\\\begin{document}$ zf'(z)/f(z) $\\\\end{document}</tex-math></inline-formula> is subordinated to <inline-formula><tex-math id=\\\"M13\\\">\\\\begin{document}$ \\\\varphi_L(z) $\\\\end{document}</tex-math></inline-formula> and these functions are related to the analytic function <inline-formula><tex-math id=\\\"M14\\\">\\\\begin{document}$ p:\\\\mathbb{D}\\\\to \\\\mathbb{C} $\\\\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\\\"M15\\\">\\\\begin{document}$ p(z)\\\\in \\\\Omega $\\\\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id=\\\"M16\\\">\\\\begin{document}$ z\\\\in \\\\mathbb{D} $\\\\end{document}</tex-math></inline-formula> by <inline-formula><tex-math id=\\\"M17\\\">\\\\begin{document}$ p(z) = zf'(z)/f(z) $\\\\end{document}</tex-math></inline-formula>. Using the admissibility criteria of the first and second order differential subordination, we investigate several subordination results for functions <inline-formula><tex-math id=\\\"M18\\\">\\\\begin{document}$ p $\\\\end{document}</tex-math></inline-formula> to satisfy <inline-formula><tex-math id=\\\"M19\\\">\\\\begin{document}$ p(z)\\\\in \\\\Omega $\\\\end{document}</tex-math></inline-formula>. As applications, we give several sufficient conditions for functions <inline-formula><tex-math id=\\\"M20\\\">\\\\begin{document}$ f $\\\\end{document}</tex-math></inline-formula> to satisfy <inline-formula><tex-math id=\\\"M21\\\">\\\\begin{document}$ zf'(z)/f(z)\\\\in \\\\Omega $\\\\end{document}</tex-math></inline-formula>.</p>\",\"PeriodicalId\":93334,\"journal\":{\"name\":\"Mathematical foundations of computing\",\"volume\":\"7 1\",\"pages\":\"591-600\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical foundations of computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/mfc.2022036\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical foundations of computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/mfc.2022036","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Analytic function that map the unit disk into the inside of the lemniscate of Bernoulli
The function \begin{document}$ \varphi_L $\end{document} defined by \begin{document}$ \varphi_L(z) = \sqrt{1+z} $\end{document} maps the unit disk \begin{document}$ \mathbb{D} $\end{document} onto \begin{document}$ \Omega = \{w\in\mathbb{C}: |w^2-1|<1\} $\end{document}, the region in the right half-plane bounded by the lemniscate of Bernoulli \begin{document}$ |w^2-1| = 1 $\end{document}. This paper deals with starlike functions defined on \begin{document}$ \mathbb{D} $\end{document} with \begin{document}$ zf'(z)/f(z)\in \Omega $\end{document} or equivalently \begin{document}$ zf'(z)/f(z) $\end{document} is subordinated to \begin{document}$ \varphi_L(z) $\end{document} and these functions are related to the analytic function \begin{document}$ p:\mathbb{D}\to \mathbb{C} $\end{document} with \begin{document}$ p(z)\in \Omega $\end{document} for all \begin{document}$ z\in \mathbb{D} $\end{document} by \begin{document}$ p(z) = zf'(z)/f(z) $\end{document}. Using the admissibility criteria of the first and second order differential subordination, we investigate several subordination results for functions \begin{document}$ p $\end{document} to satisfy \begin{document}$ p(z)\in \Omega $\end{document}. As applications, we give several sufficient conditions for functions \begin{document}$ f $\end{document} to satisfy \begin{document}$ zf'(z)/f(z)\in \Omega $\end{document}.