On certain properties of the class $U(\lambda)$

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.