Fekete-Szegö and Hankel inequalities related to the sine function

The Fekete-Szegö inequality is one of the inequalities for the coefficients which associated with the famous Bieberbach conjecture. Other issues associated with this inequality are determining the Hankel determinant denoted as Hd inequalities which are involved in the study of the singularities and integral coefficients in the Taylor series representations. In this investigation, we discuss such inequalities for certain mappings g for which (g′(z))α2zg′(z)g(z)−g(−z)(1−α)≺1+sin⁡z,(0≤α≤1)lies in the image domain starlike about 1 and symmetric about real axis. Particularly, we obtain certain inequality for functions involving Poisson distribution. Furthermore, we also find the second Hankel determinant and other inequalities related to this newly defined class.