On logarithmic coefficients of some close-to-convex functions

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 $f(0)=0=f^{\prime }(0)-1$ 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.

13 pages