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

Abstract

AbstractRecently, 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 $\mathcal{R}_{\alpha ,\Upsilon}^{\lambda ,q}(\delta ,\eta )$ R α , ϒ λ , q ( δ , η ) of analytic functions was studied: $$\begin{aligned} &\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta ) \\ &\quad := \biggl\{ \mathcal{ F}: {\Re} \biggl( (1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta}+(\delta -2\eta ) \bigl(\mathcal{H} _{\Upsilon}^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{ \prime}}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon}^{\lambda ,q} \mathcal{F}( \zeta ) \bigr) ^{{{\prime \prime}}} \biggr) \biggr\} \\ &\quad >\alpha \quad (0\leqq \alpha < 1). \end{aligned}$$ R α , ϒ λ , q ( δ , η ) : = { F : ℜ ( ( 1 − δ + 2 η ) H ϒ λ , q F ( ζ ) ζ + ( δ − 2 η ) ( H ϒ λ , q F ( ζ ) ) ′ + η ζ ( H ϒ λ , q F ( ζ ) ) ″ ) } > α ( 0 ≦ α < 1 ) . For these general analytic functions $\mathcal{F}\in \mathcal{R}_{\beta ,\Upsilon}^{\lambda ,q}(\delta , \eta )$ F ∈ R β , ϒ λ , q ( δ , η ) , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.