{"title":"Sharp Coefficient Results on the Inverse of Silverman Starlike Functions","authors":"L. Shi, M. Arif","doi":"10.3103/s1068362324700213","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In the present paper, we consider a subclass of starlike functions <span>\\(\\mathcal{G}_{\\mu}\\)</span> introduced by Silverman. It is defined by the ratio of analytic representations of convex and starlike functions. The main aim is to determine the sharp bounds of coefficient problems for the inverse of functions in this class. We derive the upper bounds of some initial coefficients, the Fekete–Szegö type inequality and the second Hankel determinant <span>\\(\\mathcal{H}_{2,2}\\left(f^{-1}\\right)\\)</span> for <span>\\(f\\in\\mathcal{G}_{\\mu}\\)</span>. On the third Hankel determinant <span>\\(\\mathcal{H}_{3,1}\\left(f^{-1}\\right)\\)</span>, we give a bound on the inverse of <span>\\(f\\in\\mathcal{G}\\)</span>. All the results are proved to be sharp.</p>","PeriodicalId":54854,"journal":{"name":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","volume":"57 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Contemporary Mathematical Analysis-Armenian Academy of Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3103/s1068362324700213","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the present paper, we consider a subclass of starlike functions \(\mathcal{G}_{\mu}\) introduced by Silverman. It is defined by the ratio of analytic representations of convex and starlike functions. The main aim is to determine the sharp bounds of coefficient problems for the inverse of functions in this class. We derive the upper bounds of some initial coefficients, the Fekete–Szegö type inequality and the second Hankel determinant \(\mathcal{H}_{2,2}\left(f^{-1}\right)\) for \(f\in\mathcal{G}_{\mu}\). On the third Hankel determinant \(\mathcal{H}_{3,1}\left(f^{-1}\right)\), we give a bound on the inverse of \(f\in\mathcal{G}\). All the results are proved to be sharp.
期刊介绍：
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) is an outlet for research stemming from the widely acclaimed Armenian school of theory of functions, this journal today continues the traditions of that school in the area of general analysis. A very prolific group of mathematicians in Yerevan contribute to this leading mathematics journal in the following fields: real and complex analysis; approximations; boundary value problems; integral and stochastic geometry; differential equations; probability; integral equations; algebra.
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