Zalcman and generalized Zalcman conjecture for the class $\mathcal{U}$

Abstract: 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.