Upper bound for the second and third Hankel determinants of analytic functions associated with the error function and q-convolution combination

Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class ${\mathcal{R}}_{\alpha ,\mathrm{\Upsilon }}^{\lambda ,q}(\delta ,\eta )$ $\mathcal{R}_{\alpha ,\Upsilon}^{\lambda ,q}(\delta ,\eta )$ of analytic functions was studied: $\begin{array}{rl}& {\mathcal{R}}_{\alpha ,\mathrm{\Upsilon }}^{\lambda ,q}(\delta ,\eta )\\ & :=\{\mathcal{F}:\mathrm{\Re }((1-\delta +2\eta )\frac{{\mathcal{H}}_{\mathrm{\Upsilon }}^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta }+(\delta -2\eta ){({\mathcal{H}}_{\mathrm{\Upsilon }}^{\lambda ,q}\mathcal{F}(\zeta ))}^{\prime }+\eta \zeta {({\mathcal{H}}_{\mathrm{\Upsilon }}^{\lambda ,q}\mathcal{F}(\zeta ))}^{″})\}\\ & >\alpha (0\leqq \alpha <1).\end{array}$ \(\begin{aligned} &\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta ) \\ &\quad := \biggl\{ \mathcal{ F}: {\Re} \biggl( (1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta}+(\delta -2\eta ) \bigl(\mathcal{H} _{\Upsilon}^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{ \prime}}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon}^{\lambda ,q} \mathcal{F}( \zeta ) \bigr) ^{{{\prime \prime}}} \biggr) \biggr\} \\ &\quad >\alpha \quad (0\leqq \alpha < 1). \end{aligned}\)

For these general analytic functions $\mathcal{F}\in {\mathcal{R}}_{\beta ,\mathrm{\Upsilon }}^{\lambda ,q}(\delta ,\eta )$ $\mathcal{F}\in \mathcal{R}_{\beta ,\Upsilon}^{\lambda ,q}(\delta , \eta )$ , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.