The estimates for the second Hankel determinant a 2 a 4 − a 3 2 of the analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ , for which either z f ′ ( z ) / f ( z ) or 1 + z f ″ ( z ) / f ′ ( z ) is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained.

Functions that are analytic in the unit disk and satisfy the differential equation f′(z) + αzf”(z) + γz²f′”(z) = g(z) are considered, where g is subordinated to a normalized convex univalent function h. These functions f are given by a double integral operator of the form with G′ subordinated to h. The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function h.

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In this manuscript, we apply the simulation mappings to present and verify several original results of common and coincidence fixed point in complete S-metric spaces. Moreover, using S-metric to expand and generalized diverse results in the literature involving simulation mappings. On the other hand, we apply our major results to derive several common and coincidencefixed point theorems for right monotone simulation map in complete S-metric. As implementations, various related outcomes of fixed-point theory via specific simulation mappings are obtained in complete S-metric spaces. Additionally, illustrative examples and some applications to solve an integral equation are introduced to support our major results.