The function $f(z)=z+\Sigma^{\infty}_{n=2}a_nz^n$, normalized, analytic and univalent in the unit disk $\mathbb D=\{z:|z|<1\}$, belongs to the class $\mathcal U$ if, and only if, \[ \Big|\Big(\frac z{f(z)}\Big)^2f'(z)-1\Big|<1\qquad(zı\mathbb D). \] In this paper we prove the Zalcman and the generalized Zalcman conjecture for the class U and some values of parameters in the conjectures.