bound associated with certain hankel determinants and zalcman conjecture for multivalent functions of bounded turning

In this paper, we investigate for a sharp upper bound to certain generalized second hankel determinant, the zalcman conjecture and an upper bound for the third, fourth hankel determinants for the class of multivalent analytic bounded turning functions. further, we estimate an upper bound for third and fourth hankel determinants with respect to two-fold and three-fold symmetric functions belongs to the same class. the practical tools applied in deriving of our main results are the coefficient inequalities of the carath´eodory class p.