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On logarithmic coefficients of some close-to-convex functions


Authors: Md Firoz Ali and A. Vasudevarao
Journal: Proc. Amer. Math. Soc. 146 (2018), 1131-1142
MSC (2010): Primary 30C45, 30C55
DOI: https://doi.org/10.1090/proc/13817
Published electronically: October 5, 2017
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Abstract: The logarithmic coefficients $ \gamma _n$ of an analytic and univalent function $ f$ in the unit disk $ \mathbb{D}=\{z\in \mathbb{C}:\vert z\vert<1\}$ with the normalization $ f(0)=0=f'(0)-1$ are defined by $ \log \frac {f(z)}{z}= 2\sum _{n=1}^{\infty } \gamma _n z^n$. Recently, D. K. Thomas [Proc. Amer. Math. Soc. 144 (2016), 1681-1687] proved that $ \vert\gamma _3\vert\le \frac {7}{12}$ for functions in a subclass of close-to-convex functions (with argument 0) and claimed that the estimate is sharp by providing a form of an extremal function. In the present paper, we point out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument 0). We also determine a sharp upper bound of $ \vert\gamma _3\vert$ for close-to-convex functions (with argument 0) with respect to the Koebe function.


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Additional Information

Md Firoz Ali
Affiliation: Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721 302, West Bengal, India
Email: ali.firoz89@gmail.com

A. Vasudevarao
Affiliation: Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721 302, West Bengal, India
Email: alluvasu@maths.iitkgp.ernet.in

DOI: https://doi.org/10.1090/proc/13817
Keywords: Univalent, starlike, convex, close-to-convex functions, logarithmic coefficient.
Received by editor(s): June 16, 2016
Received by editor(s) in revised form: April 14, 2017
Published electronically: October 5, 2017
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2017 American Mathematical Society

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