Some inequalities for a certain subclass of starlike functions

In 2011, Sok'{o}{\l} (Comput. Math. Appl. 62, 611–619) introduced and studied the class $\mathcal{S}\mathcal{K}(\alpha )$ as a certain subclass of starlike functions, consists of all functions $f$ ($f(0)=0=f^{\mathrm{\prime }}(0)-1$) 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 $-3<\alpha \le 1$. Also, he obtained some interesting results for the class $\mathcal{S}\mathcal{K}(\alpha )$. In this paper, some another properties of this class, including infimum of $\mathfrak{R}\mathfrak{e}\frac{f(z)}{z}$, order of strongly starlikeness, the sharp logarithmic coefficients inequality and the sharp Fekete-Szeg"{o} inequality are investigated.