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.
Categories: math.CV