On a certain subclass of strongly starlike functions

Let $\mathcal{S}^*(\alpha_1,\alpha_2)$, where $\alpha_1, \alpha_2 \in (0,1]$, represent the class of functions $f$ that are analytic in the open unit disk $\mathbb{D}$, normalized by $f(0) = f'(0) - 1=0$, and satisfying the following double-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\mathbb{D}). \end{equation*} In this manuscript, we estimate the coefficients and logarithmic coefficients associated with functions that belong to the class $\mathcal{S}^*(\alpha_1,\alpha_2)$. As a result, we provide a general bound for the coefficients of a strongly starlike function, which has been an open question until now. Finally, we derive upper and lower bounds for the expression ${\rm Re}\{zf'(z)/f(z)\}$, where $f\in \mathcal{S}^*(\alpha_1,\alpha_2)$.