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On the generalized Zalcman functional $ \lambda a_n^2-a_{2n-1}$ in the close-to-convex family


Authors: Liulan Li and Saminathan Ponnusamy
Journal: Proc. Amer. Math. Soc. 145 (2017), 833-846
MSC (2010): Primary 30C45; Secondary 30C20, 30C55
DOI: https://doi.org/10.1090/proc/13260
Published electronically: August 23, 2016
MathSciNet review: 3577882
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Abstract: Let $ {\mathcal S}$ denote the class of all functions $ f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ analytic and univalent in the unit disk $ \mathbb{D}$. For $ f\in {\mathcal S}$, Zalcman conjectured that $ \vert a_n^2-a_{2n-1}\vert\leq (n-1)^2$ for $ n\geq 3$. This conjecture has been verified for only certain values of $ n$ for $ f\in {\mathcal S}$ and for all $ n\ge 4$ for the class $ \mathcal C$ of close-to-convex functions (and also for a couple of other classes). In this paper we provide bounds of the generalized Zalcman coefficient functional $ \vert\lambda a_n^2-a_{2n-1}\vert$ for functions in $ \mathcal C$ and for all $ n\ge 3$, where $ \lambda $ is a positive constant.


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Additional Information

Liulan Li
Affiliation: College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan 421002, People’s Republic of China
Email: lanlimail2012@sina.cn

Saminathan Ponnusamy
Affiliation: Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India
Email: samy@isichennai.res.in, samy@iitm.ac.in

DOI: https://doi.org/10.1090/proc/13260
Keywords: Univalent, convex, starlike and close-to-convex functions, Fekete-Szeg\"o inequality, Zalcman and generalized Zalcman functionals
Received by editor(s): February 15, 2016
Received by editor(s) in revised form: April 23, 2016
Published electronically: August 23, 2016
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society