On the generalized Zalcman functional $\lambda a_n^2-a_{2n-1}$ in the close-to-convex family
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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 $|a_n^2-a_{2n-1}|\leq (n-1)^2$ 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 $|\lambda a_n^2-a_{2n-1}|$ for functions in $\mathcal C$ and for all $n\ge 3$, where $\lambda$ is a positive constant.References
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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
- MR Author ID: 771284
- 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
- MR Author ID: 259376
- ORCID: 0000-0002-3699-2713
- Email: samy@isichennai.res.in, samy@iitm.ac.in
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 833-846
- MSC (2010): Primary 30C45; Secondary 30C20, 30C55
- DOI: https://doi.org/10.1090/proc/13260
- MathSciNet review: 3577882