Let ${\mathcal{S}}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ which are of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ . We determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in ${\mathcal{S}}$ and certain of its subclasses. We also discuss similar problems for typically real functions.

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 $f(0)=0=f’(0)-1$ are defined by $\log \frac {f(z)}{z}= 2\sum _{n=1}^{\infty } \gamma _n z^n$ . Recently, D. K. Thomas [Proc. Amer. Math. Soc. 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 an extremal function. In the present paper, we point 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.