  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
$f(z)=z+a_2z^2+\cdots$.
