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

TL;DR

This paper investigates the properties of the class al() of analytic functions in the unit disk, exploring conjectures and inequalities related to this class, including the Zalcman conjecture and Hankel determinants.

Contribution

It introduces new results on the properties of the class al(), including proofs and bounds for conjectures and inequalities for functions within this class.

Findings

Results on the Zalcman conjecture for al()

Bounds on the second and third order Hankel determinants

Verification of the Krushkal inequality within the class

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 $f(z)=z+a_2z^2+a_3z^3+\cdots$. 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.