On logarithmic coefficients of some close-to-convex functions

TL;DR

This paper refines the bounds on the third logarithmic coefficient for close-to-convex functions, correcting previous claims of sharpness and providing more accurate estimates.

Contribution

It disproves the existence of extremal functions claimed earlier and establishes a sharper upper bound for the third logarithmic coefficient.

Findings

Previous bounds on |b3_3| are not sharp.

New upper bound for |b3_3| is established.

The bound is sharp with respect to the Koebe function.

Abstract

The logarithmic coefficients $\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty} \gamma_n z^n$. 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 $|\gamma_3|\le \frac{7}{12}$ 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 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 $|\gamma_3|$ for close-to-convex functions (with argument $0$) with respect to the Koebe function.