On logarithmic coefficients of some close-to-convex functions

2016/06/16 by Md Firoz Ali, Ali, Md Firoz, A. Vasudevarao +1
Mathematics · #30C45 #30C55 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1606.05162

openalex publication_date 2016/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The logarithmic coefficients γ_n of an analytic and univalent function f in the unit disk \mathbbD=\z∈ℂ:|z|<1\ with the normalization f(0)=0=f'(0)-1 is defined by log (f(z))/(z)= 2∑_n=1^∞ γ_n zⁿ. Recently, D.K. Thomas [On the logarithmic coefficients of close to convex functions, \it Proc. Amer. Math. Soc. \bf 144 (2016), 1681--1687] proved that |γ₃|≤ (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 a extremal function. In the present paper, we pointed 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 |γ₃| for close-to-convex functions (with argument 0) with respect to the Koebe function.

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