  In 2011, Sok\'{o}{\l} (Comput. Math. Appl. 62, 611--619) introduced and
studied the class $\mathcal{SK}(\alpha)$ as a certain subclass of starlike
functions, consists of all functions $f$ ($f(0)=0=f'(0)-1$) which satisfy in
the following subordination relation: \begin{equation*}
\frac{zf'(z)}{f(z)}\prec \frac{3}{3+(\alpha-3)z-\alpha z^2} \qquad |z|<1,
\end{equation*} where $-3<\alpha\leq1$. Also, he obtained some interesting
results for the class $\mathcal{SK}(\alpha)$. In this paper, some another
properties of this class, including infimum of $\mathfrak{Re}\frac{f(z)}{z}$,
order of strongly starlikeness, the sharp logarithmic coefficients inequality
and the sharp Fekete-Szeg\"{o} inequality are investigated.
