{"title":"某些等价函数类中反函数对数系数的第二汉克尔行列式","authors":"Sanju Mandal, Molla Basir Ahamed","doi":"10.1007/s10986-024-09623-5","DOIUrl":null,"url":null,"abstract":"<p>The Hankel determinant <span>\\({H}_{\\mathrm{2,1}}\\left({F}_{f-1}/2\\right)\\)</span> of logarithmic coefficients is defined as</p><p><span>\\({H}_{\\mathrm{2,1}}\\left({F}_{f-1}/2\\right):=\\left|\\begin{array}{cc}{\\Gamma }_{1}& {\\Gamma }_{2}\\\\ {\\Gamma }_{2}& {\\Gamma }_{3}\\end{array}\\right|={\\Gamma }_{1}{\\Gamma }_{3}-{\\Gamma }_{2}^{2},\\)</span></p><p>where <span>\\({\\Gamma }_{1},{\\Gamma }_{2},\\)</span> and <span>\\({\\Gamma }_{3}\\)</span> are the first, second, and third logarithmic coefficients of inverse functions belonging to the class <span>\\(\\mathcal{S}\\)</span> of normalized univalent functions. In this paper, we establish sharp inequalities <span>\\(\\left|{H}_{\\mathrm{2,1}}\\left({F}_{f-1}/2\\right)\\right|\\le 19/288,\\)</span> <span>\\(\\left|{H}_{\\mathrm{2,1}}\\left({F}_{f-1}/2\\right)\\right|\\le 1/144,\\)</span> and <span>\\(\\left|{H}_{\\mathrm{2,1}}\\left({F}_{f-1}/2\\right)\\right|\\le 1/36\\)</span> for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions\",\"authors\":\"Sanju Mandal, Molla Basir Ahamed\",\"doi\":\"10.1007/s10986-024-09623-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Hankel determinant <span>\\\\({H}_{\\\\mathrm{2,1}}\\\\left({F}_{f-1}/2\\\\right)\\\\)</span> of logarithmic coefficients is defined as</p><p><span>\\\\({H}_{\\\\mathrm{2,1}}\\\\left({F}_{f-1}/2\\\\right):=\\\\left|\\\\begin{array}{cc}{\\\\Gamma }_{1}& {\\\\Gamma }_{2}\\\\\\\\ {\\\\Gamma }_{2}& {\\\\Gamma }_{3}\\\\end{array}\\\\right|={\\\\Gamma }_{1}{\\\\Gamma }_{3}-{\\\\Gamma }_{2}^{2},\\\\)</span></p><p>where <span>\\\\({\\\\Gamma }_{1},{\\\\Gamma }_{2},\\\\)</span> and <span>\\\\({\\\\Gamma }_{3}\\\\)</span> are the first, second, and third logarithmic coefficients of inverse functions belonging to the class <span>\\\\(\\\\mathcal{S}\\\\)</span> of normalized univalent functions. In this paper, we establish sharp inequalities <span>\\\\(\\\\left|{H}_{\\\\mathrm{2,1}}\\\\left({F}_{f-1}/2\\\\right)\\\\right|\\\\le 19/288,\\\\)</span> <span>\\\\(\\\\left|{H}_{\\\\mathrm{2,1}}\\\\left({F}_{f-1}/2\\\\right)\\\\right|\\\\le 1/144,\\\\)</span> and <span>\\\\(\\\\left|{H}_{\\\\mathrm{2,1}}\\\\left({F}_{f-1}/2\\\\right)\\\\right|\\\\le 1/36\\\\)</span> for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-02-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10986-024-09623-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10986-024-09623-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
where \({\Gamma }_{1},{\Gamma }_{2},\) and \({\Gamma }_{3}\) are the first, second, and third logarithmic coefficients of inverse functions belonging to the class \(\mathcal{S}\) of normalized univalent functions. In this paper, we establish sharp inequalities \(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 19/288,\)\(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 1/144,\) and \(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 1/36\) for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively.