Sharp bounds associated with the Zalcman conjecture for the initial coefficients and second Hankel determinants for certain subclass of analytic functions

In this paper, we obtain sharp bounds in the Zalcman conjecture for the initial coe cients, the second Hankel determinant $H_{2,2}(f) = a_2a_4 - a^2_3$ and an upper bound for the second Hankel determinant $H_{2,3}(f) = a_3a_5-a_2$ for the functions belonging to a certain subclass of analytic functions. The practical tools applied in the derivation of our main results are the coe cient inequalities of the Caratheodory class $P$.