Bounds of Hankel determinants for analytic function

Abstract

In this paper, we give estimates of the Hankel determinant $H_{2}(1)$ in a novel class $\mathcal{N}\left( \varepsilon \right) $ of analytical functions in the unit disc. In addition, the relation between the Fekete-Szego function $H_{2}(1)$ and the module of the angular derivative of the analytical function $p(z)$ at a boundary point $b$ of the unit disk will be given. In this association, the coefficients in the Hankel determinant $b_{2}$, $b_{3}$ and $b_{4}$ will be taken into consideration. Moreover, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.