On coefficient problems for close-to-star functions

Let ${\mathcal A}$ be the class of analytic functions in the unit disk $\mathbb{D}$ which have the form $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$. And let ${\mathcal{CST}}$ be the subclass ${\mathcal A}$ consisting close-to-star functions. For given $q$, $n \in \mathbb{N}$ and $f\in{\mathcal A}$, the Hankel determinants $H_{q,n}(f)$ is defined as
\begin{equation*} H_{q,n}(f) := \begin{vmatrix} a_{n} &a_{n+1} &\cdots &a_{n+q-1} \\ a_{n+1} &a_{n+2} &\cdots &a_{n+q} \\ \vdots &\vdots &\vdots &\vdots \\ a_{n+q-1} &a_{n+q} &\cdots &a_{n+2(q-1)} \end{vmatrix}. \end{equation*}
And, for given $m$, $n \in \mathbb{N}\setminus\{1\}$, the Zalcman functional $J_{n,m}(f)$ of $f\in{\mathcal A}$ is defined by
$$ J_{n,m}(f):=a_{n+m-1} -a_n a_m. $$
In this talk, we discuss the sharp estimates of the second Hankel determinants such as $H_{2,1}$ and $H_{2,2}$ and the Zalcman functional $J_{2,3}$ over several subclasses of ${\mathcal{CST}}$. Also, the sharp bounds of early logarithmic coefficients and coefficients of the inverses of close-to-star functions are investigated.
This is a joint work with Oh Sang Kwon (Kyungsung University, Busan, Korea) and Nak Eun Cho (Pukyong National University, Busan, Korea).