Deekonda Vamshee Krishna - Department of Mathematics, North-Eastern Hill University (NEHU), Umshing Mawkynroh, Meghalaya-۷۹۳۰۲۲, Shillong, Bharat.
Nallamothu Vani - Department of Mathematics, GITAM School of Science, GITAM University, Andhra Pradesh-۵۳۰۰۴۵, Visakhapatnam, India.
Biswajit Rath - Department of Mathematics, GITAM School of Science, GITAM University, Andhra Pradesh-۵۳۰۰۴۵, Visakhapatnam, India.
Karri Sanjay Kumar - Department of Mathematics, GITAM School of Science, GITAM University, Andhra Pradesh-۵۳۰۰۴۵, Visakhapatnam, India.

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\acute{e}odory class \mathcal{P}.

p-valent bounded turning function, Upper bound, Hankel determinant, Univalent function, Carath´eodory function, Hypergeometric Function