Results Related to the Sharp Coefficients Bounds for Symmetric Convex Functions Connected With the Sigmoid Function

A challenging task in studying geometric function theory is determining the sharp estimates for coefficient-related results that come up in the Taylor series of analytic univalent functions. In the current paper, we attempt to establish some estimates of results concerning coefficient inequalities for the class of convex functions with respect to symmetric points and connected with the sigmoid function. These results involve the Kruskal and Zalcman inequalities as well as the Hankel determinant of Order two. Furthermore, the logarithmic coefficients play a vital role in the results of the univalent function theory. We achieve the estimates of the initial logarithmic coefficients, the Kruskal and Zalcman inequalities, as well as the second Hankel determinant of logarithmic coefficients for the defined class. Moreover, determining the estimates for the inverse function is a more challenging task than estimating the function itself. The estimates of the initial logarithmic coefficients, the Fekete–Szegö inequality, and the second Hankel determinant of inverse coefficients for the abovementioned class are also investigated. The estimates in this paper have shown to be sharp.