  Function $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$, normalized, analytic and
univalent in the unit disk $\mathbb D=\{z:|z|<1\}$, belongs to the class
$\mathcal{U}$. if, and only if, \[ \left| \left(\frac{z}{f(z)}\right)^2
-1\right|<1 \quad\quad (z\in \mathbb D). \] In this paper, we prove the Zalcman
and the generalized Zalcman conjecture for the class $\mathcal{U}$ and some
values of parameters in the conjectures.
