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 $f(0)=0=f’(0)-1$ 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.