{"title":"On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis","authors":"P. Zaprawa","doi":"10.5802/CRMATH.144","DOIUrl":null,"url":null,"abstract":"In this paper we consider two functionals of the Fekete–Szegö type: Φ f (μ) = a2a4 − μa3 and Θ f (μ) = a4 −μa2a3 for analytic functions f (z) = z +a2z +a3z + . . ., z ∈∆, (∆= {z ∈C : |z| < 1}) and for real numbers μ. For f which is univalent and convex in the direction of the imaginary axis, we find sharp bounds of the functionals Φ f (μ) and Θ f (μ). It is possible to transfer the results onto the class KR(i ) of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class T of typically real functions. As corollaries, we obtain bounds of the second Hankel determinant in KR(i ) and T . 2020 Mathematics Subject Classification. 30C50. Funding. The project/research was financed in the framework of the project Lublin University of Technology Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract no.030/RID/2018/19). Manuscript received 14th August 2019, revised 24th June 2020, accepted 3rd November 2020.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"1 1","pages":"1213-1226"},"PeriodicalIF":0.8000,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Comptes Rendus Mathematique","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.5802/CRMATH.144","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
In this paper we consider two functionals of the Fekete–Szegö type: Φ f (μ) = a2a4 − μa3 and Θ f (μ) = a4 −μa2a3 for analytic functions f (z) = z +a2z +a3z + . . ., z ∈∆, (∆= {z ∈C : |z| < 1}) and for real numbers μ. For f which is univalent and convex in the direction of the imaginary axis, we find sharp bounds of the functionals Φ f (μ) and Θ f (μ). It is possible to transfer the results onto the class KR(i ) of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class T of typically real functions. As corollaries, we obtain bounds of the second Hankel determinant in KR(i ) and T . 2020 Mathematics Subject Classification. 30C50. Funding. The project/research was financed in the framework of the project Lublin University of Technology Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract no.030/RID/2018/19). Manuscript received 14th August 2019, revised 24th June 2020, accepted 3rd November 2020.
本文考虑了两个Fekete-Szegö型泛函:Φ f (μ) = a2a4−μ a3和Θ f (μ) = a4−μa2a3,适用于解析函数f (z) = z +a2z +a3z +…,z∈∆,(∆= {z∈C: |z| < 1})和实数μ。对于虚轴方向上的一元凸函数f,我们得到了函数Φ f (μ)和Θ f (μ)的明确界限。可以将结果转移到KR(i)类的虚轴方向凸函数与实系数,也可以转移到T类的典型实数函数。作为推论,我们得到了在KR(i)和T中的第二汉克尔行列式的界。2020数学学科分类。30C50。资金。该项目/研究由波兰科学和高等教育部资助的卢布林科技大学区域卓越倡议项目(合同编号030/RID/2018/19)框架内资助。收稿于2019年8月14日,改稿于2020年6月24日,收稿于2020年11月3日。
期刊介绍:
The Comptes Rendus - Mathématique cover all fields of the discipline: Logic, Combinatorics, Number Theory, Group Theory, Mathematical Analysis, (Partial) Differential Equations, Geometry, Topology, Dynamical systems, Mathematical Physics, Mathematical Problems in Mechanics, Signal Theory, Mathematical Economics, …
Articles are original notes that briefly describe an important discovery or result. The articles are written in French or English.
The journal also publishes review papers, thematic issues and texts reflecting the activity of Académie des sciences in the field of Mathematics.