{"title":"解析函数子类的汉克尔和托普利兹行列式","authors":"V. MatematychniStudii., No 60, M. Buyankara, M.","doi":"10.30970/ms.60.2.132-137","DOIUrl":null,"url":null,"abstract":"Let the function $f\\left( z \\right) =z+\\sum_{k=2}^{\\infty}a{_{k}}z {^{k}}\\in A$ be locally univalent for $z \\in \\mathbb{D}%:=\\{z \\in \\mathbb{C}:{|}z {|}<1\\}$ and $0\\leq\\alpha<1$.Then, $f$\\textit{\\ }$\\in $ $M(\\alpha )$ if and only if \\begin{equation*}\\Re\\Big( \\left( 1-z ^{2}\\right) \\frac{f(z )}{z }\\Big) >\\alpha,\\quad z \\in \\mathbb{D}.\\end{equation*}%Due to their geometrical characteristics, this class has a significantimpact on the theory of geometric functions. In the article we obtain sharp bounds for the second Hankel determinant \\begin{equation*}\\left\\vert H_{2}\\left( 2\\right) \\left( f\\right) \\right\\vert =\\left\\verta_{2}a_{4}-{a_{3}^{2}}\\right\\vert \\end{equation*}and some Toeplitz determinants \\begin{equation*}\\left\\vert {T}_{3}\\left( 1\\right) \\left( f\\right) \\right\\vert =\\left\\vert 1-2%{a_{2}^{2}}+2{a_{2}^{2}}a_{3}-{a_{3}^{2}}\\right\\vert,\\ \\\\left\\vert {T}_{3}\\left( 2\\right) \\left( f\\right) \\right\\vert =\\left\\vert {%a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}}\\right\\vert \\end{equation*}of a subclass of analytic functions $M(\\alpha )$ in the open unit disk $%\\mathbb{D}$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hankel and Toeplitz determinants for a subclass of analytic functions\",\"authors\":\"V. MatematychniStudii., No 60, M. Buyankara, M.\",\"doi\":\"10.30970/ms.60.2.132-137\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let the function $f\\\\left( z \\\\right) =z+\\\\sum_{k=2}^{\\\\infty}a{_{k}}z {^{k}}\\\\in A$ be locally univalent for $z \\\\in \\\\mathbb{D}%:=\\\\{z \\\\in \\\\mathbb{C}:{|}z {|}<1\\\\}$ and $0\\\\leq\\\\alpha<1$.Then, $f$\\\\textit{\\\\ }$\\\\in $ $M(\\\\alpha )$ if and only if \\\\begin{equation*}\\\\Re\\\\Big( \\\\left( 1-z ^{2}\\\\right) \\\\frac{f(z )}{z }\\\\Big) >\\\\alpha,\\\\quad z \\\\in \\\\mathbb{D}.\\\\end{equation*}%Due to their geometrical characteristics, this class has a significantimpact on the theory of geometric functions. In the article we obtain sharp bounds for the second Hankel determinant \\\\begin{equation*}\\\\left\\\\vert H_{2}\\\\left( 2\\\\right) \\\\left( f\\\\right) \\\\right\\\\vert =\\\\left\\\\verta_{2}a_{4}-{a_{3}^{2}}\\\\right\\\\vert \\\\end{equation*}and some Toeplitz determinants \\\\begin{equation*}\\\\left\\\\vert {T}_{3}\\\\left( 1\\\\right) \\\\left( f\\\\right) \\\\right\\\\vert =\\\\left\\\\vert 1-2%{a_{2}^{2}}+2{a_{2}^{2}}a_{3}-{a_{3}^{2}}\\\\right\\\\vert,\\\\ \\\\\\\\left\\\\vert {T}_{3}\\\\left( 2\\\\right) \\\\left( f\\\\right) \\\\right\\\\vert =\\\\left\\\\vert {%a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}}\\\\right\\\\vert \\\\end{equation*}of a subclass of analytic functions $M(\\\\alpha )$ in the open unit disk $%\\\\mathbb{D}$.\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-12-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Matematychni Studii\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30970/ms.60.2.132-137\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.60.2.132-137","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
Hankel and Toeplitz determinants for a subclass of analytic functions
Let the function $f\left( z \right) =z+\sum_{k=2}^{\infty}a{_{k}}z {^{k}}\in A$ be locally univalent for $z \in \mathbb{D}%:=\{z \in \mathbb{C}:{|}z {|}<1\}$ and $0\leq\alpha<1$.Then, $f$\textit{\ }$\in $ $M(\alpha )$ if and only if \begin{equation*}\Re\Big( \left( 1-z ^{2}\right) \frac{f(z )}{z }\Big) >\alpha,\quad z \in \mathbb{D}.\end{equation*}%Due to their geometrical characteristics, this class has a significantimpact on the theory of geometric functions. In the article we obtain sharp bounds for the second Hankel determinant \begin{equation*}\left\vert H_{2}\left( 2\right) \left( f\right) \right\vert =\left\verta_{2}a_{4}-{a_{3}^{2}}\right\vert \end{equation*}and some Toeplitz determinants \begin{equation*}\left\vert {T}_{3}\left( 1\right) \left( f\right) \right\vert =\left\vert 1-2%{a_{2}^{2}}+2{a_{2}^{2}}a_{3}-{a_{3}^{2}}\right\vert,\ \\left\vert {T}_{3}\left( 2\right) \left( f\right) \right\vert =\left\vert {%a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}}\right\vert \end{equation*}of a subclass of analytic functions $M(\alpha )$ in the open unit disk $%\mathbb{D}$.