Some properties of the class $\mathcal{U}$

In this paper we study the class $\mathcal{U}$ of functions that are analytic in the open unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$ and satisfy $$\left|\left [\frac{z}{f(z)} \right]^{2}f'(z)-1 \right|<1\quad\quad (z\in {\mathbb D}).$$ For functions in the class $\mathcal{U}$ we give sharp estimate of the second ant the third Hankel determinant, its relationship with the class of $\alpha$-convex functions, as well as certain starlike properties.