On a certain subclass of strongly starlike functions

Let ${\mathcal{S}}^{\ast }({\alpha }_1,{\alpha }_2)$, 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 $f(0)=f^{\mathrm{\prime }}(0)-1=0$, 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 ${\mathcal{S}}^{\ast }({\alpha }_1,{\alpha }_2)$. 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 $\mathrm{R}\mathrm{e}\{z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\}$, where $f\in {\mathcal{S}}^{\ast }({\alpha }_1,{\alpha }_2)$.