On the Upper bounds of Hankel Determinant of $q$-Starlike Functions Linked to the Bernoulli Lemniscate

Abstract

Let $\mathcal{SL}_{q}^{*}(\varpi,\varphi)$ denote the subclass of analytic functions  defined in the open unit disk $\mathbb{U}=\{t\in\mathbb{C}:|t|<1\}$, normalized by  $h(0)=0$ and $h'(0)=1$, which satisfy certain subordination conditions \begin{align*} \frac{t \, D_q \big(\mathcal{R}_{q}^{\varpi}(h(t))\big)}{\mathcal{R}_{q}^{\varpi}(h(t))} \prec {\left(  \frac{2(1+t)}{2+(1-q)t}\right)^\frac{1}{2} }, \end{align*} where $\prec$ is the subordination relation. This class serves as a $q$-analogue of starlike functions associated with the parameter $\varpi$. In this paper, we investigate coefficient-related  problems for functions in $\mathcal{SL}_{q}^{*}(\varpi,\varphi)$. In particular, we  derive sharp bounds for the initial Taylor coefficients, the Fekete--Szeg\"{o} functional,  and the second Hankel determinant.