On a certain subclass of strongly starlike functions

Abstract

Let ${\mathcal{S}}^{\ast }(\hat{I}{\pm }_1,\hat{I}{\pm }_2)$, where $\hat{I}{\pm }_1,\hat{I}{\pm }_2\in (0,1]$, represent the class of functions $f$ that are analytic in the open unit disk $\mathbb{D}$, normalized by $f(0)=f^{\prime }(0)-1=0$, and satisfying the following double-sided inequality: \begin{equation*} -\frac{πα_1}{2}< \arg\left\{\frac{zf'(z)}{f(z)}\right\} <\frac{πα_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 ${\mathcal{S}}^{\ast }(\hat{I}{\pm }_1,\hat{I}{\pm }_2)$. 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 $\mathrm{Re}\{z{f}^{\prime }(z)/f(z)\}$, where $f\in {\mathcal{S}}^{\ast }(\hat{I}{\pm }_1,\hat{I}{\pm }_2)$.

16 pages, 1 figure