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: \(\begin{aligned} -\frac{\pi \alpha _1}{2}< \arg \left\{ \frac{zf'(z)}{f(z)}\right\} <\frac{\pi \alpha _2}{2}, \quad (z\in \mathbb {D}). \end{aligned}\) 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 \(\textrm{Re}\{zf'(z)/f(z)\}\) , where \(f\in \mathcal {S}^*(\alpha _1,\alpha _2)\) .
