  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.
