  Let $\mathcal{S}^*_t(\alpha_1,\alpha_2)$ denote the class of functions $f$
analytic in the open unit disc $\Delta$, normalized by the condition
$f(0)=0=f'(0)-1$ and satisfying the following two--sided inequality:
\begin{equation*}
  -\frac{\pi\alpha_1}{2}< \arg\left\{\frac{zf'(z)}{f(z)}\right\}
  <\frac{\pi\alpha_2}{2} \quad (z\in\Delta), \end{equation*} where
$0<\alpha_1,\alpha_2\leq1$. The class $\mathcal{S}^*_t(\alpha_1,\alpha_2)$ is a
subclass of strongly starlike functions of order $\beta$ where
$\beta=\max\{\alpha_1,\alpha_2\}$. The object of the present paper is to derive
some certain inequalities including (for example), upper and lower bounds for
${\rm Re}\{zf'(z)/f(z)\}$, growth theorem, logarithmic coefficient estimates
and coefficient estimates for functions $f$ belonging to the class
$\mathcal{S}^*_t(\alpha_1,\alpha_2)$.
