Sharp second-order Hankel determinant bounds for alpha-convex functions connected with modified sigmoid function

he study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of {normalized} univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a family of alpha-convex functions associated with modified sigmoid functions and then to investigate sharp bounds of initial coefficients, Fekete-Szeg\"{o} inequality, and second-order Hankel determinants. Moreover, we also examine the logarithmic and inverse coefficients of functions within a defined family regarding recent issues. All of the estimations that were found are sharp.