Coefficient Estimates, the Fekete–Szegö Inequality, and Hankel Determinants for Universally Prestarlike Functions Defined by Fractional Derivative in a Shell-Shaped Region

In this paper, we introduce and investigate a new subclass ${\left({\mathcal{R}}_{\varsigma }^u\right)}^{\mathfrak{g}}\left(\varphi \right)$ of universally prestarlike generalized functions of order $\varsigma$, where $\varsigma \le 1$, associated with a shell-shaped region defined by $\Lambda =\mathrm{C}\setminus [1,\infty )$ for the present investigation, by utilizing the Srivastava–Owa fractional derivative of order $\delta$. Coefficient inequalities for $|{\mathfrak{a}}_2|$ and $|{\mathfrak{a}}_3|$ for functions belonging to the newly introduced class are obtained. Additionally, the Fekete–Szegö inequality is investigated for this class of functions. In order to enhance the coefficient studies for this class, the second Hankel determinant is also evaluated.