Sharp bounds associated with the Zalcman conjecture for the initial coefficients and second Hankel determinants for certain subclass of analytic functions
Abstract:
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 $P$.