Fourth order Hankel determinants for certain subclasses of modified sigmoid-activated analytic functions involving the trigonometric sine function

The aim of this paper is to introduce two new subclasses ${\mathcal{R}}_{sin}^m(\mathrm{\Im })$ $\mathcal{R}_{\sin }^{m}(\Im )$ and ${\mathcal{R}}_{sin}(\mathrm{\Im })$ $\mathcal{R}_{\sin }(\Im )$ of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function $\mathrm{\Im }(v)=\frac{2}{1+e^{-v}}$ $\Im (v)=\frac{2}{1+e^{-v}}$ , $v\ge 0$ $v\geq 0$ in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results.