Coefficient bound associated with certain Hankel determinants and Zalcman conjecturefor a subfamily of multivalent bounded turning functions

In this paper, we introduce certain subfamily of $p$-valent analytic functions of bounded turning for which we estimate best possible upper bound to certain generalised second Hankel determinant, the Zalcman conjecture and an upper bound to the third, fourth Hankel determinants. Further, we investigate an upper bound for third and fourth Hankel determinants with respect to two-fold and three-fold symmetric functions for the same class. The practical tools applied in the derivation of our main results are the coefficient inequalities of the Carathéodory class $\mathcal{P}$.