Using the Lebedev–Milin inequalities, bounds on the logarithmic coefficients of an analytic function can be transferred to estimates on coefficients of the function itself and related functions. From this fact, the study of logarithmic-related problems of a certain subclass of univalent functions has attracted much attention in recent years. In our present investigation, a subclass of starlike functions ${\mathcal{S}}_e^{*}$ connected with the exponential mapping was considered. The main purpose of this article is to obtain the sharp estimates of the second Hankel determinant with the logarithmic coefficient as entry for this class.

In this paper, we determine the sharp upper bounds of Hankel determinants for logarithmic and inverse functions of bounded turning functions associated with the hyperbolic tangent function.

In our present investigation, a subclass of starlike function ${\mathcal{S}}_{n-1,\mathcal{L}}^{*}$ connected with a domain bounded by an epicycloid with $n-1$ cusps was considered. The main work is to investigate some coefficient inequalities, and second and third Hankel determinants for functions belonging to this class. In particular, we calculate the sharp bounds of the third Hankel determinant for $f\in {\mathcal{S}}_{4\mathcal{L}}^{*}$ with $\frac{z{f}^{\prime }(z)}{f(z)}$ bounded by a four-leaf shaped domain under the unit disk $\mathbb{D}$.