Two types of the second Hankel determinant for the class $\mathcal{U}$ and the general class $\mathcal{S}$

Abstract

In this paper we determine the upper bounds of the Hankel determinants of special type $H_2(3)(f)$ and $H_2(4)(f)$ for the class of univalent functions and for the class $\mathcal{U}$ defined by \[ \mathcal{U}=\left\{ f\in\mathcal{A} : \left|\left[\frac{z}{f(z)}\right]^2 f'(z)-1 \right|<1,\, z\in{\mathbb D} \right\}, \] where $\mathcal{A}$ is the class of functions analytic in the unit disk $\mathbb{D}$ and normalized such that $f(z)=z+a_2{z}^2+\cdots$.