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

Abstract

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 $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots$. In this paper we study the class $\mathcal{U}(\lambda )$, $0<\lambda \le 1$, consisting of functions $f$ from $\mathcal{A}$ satisfying \[\left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right| < λ\quad (z\in {\mathbb D}).\] and give results regarding the Zalcman Conjecture, the generalised Zalcman conjecture, the Krushkal inequality and the second and third order Hankel determinant.