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

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 $$\mathcal{U}=\{f\in \mathcal{A}:\mid {\left[\frac{z}{f(z)}\right]}^2{f}^{\mathrm{\prime }}(z)-1\mid <1,z\in \mathbb{D}\},$$ where $\mathcal{A}$ is the class of functions analytic in the unit disk $\mathbb{D}$ and normalized such that $f(z)=z+a_2{z}^2+\cdots$.