  Let ${\mathcal A}$ be the class of functions analytic in the unit disk
${\mathbb D} := \{ z\in {\mathbb C}:\, |z| < 1 \}$ and normalized such that
$f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we study the class
$\mathcal{U}(\lambda)$, $0<\lambda \leq1$, consisting of functions $f$ from
${\mathcal{A}}$ satisfying \[\left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right|
< \lambda \quad (z\in {\mathbb D}).\] and give results regarding the Zalcman
Conjecture, the generalised Zalcman conjecture, the Krushkal inequality and the
second and third order Hankel determinant.
