Two types of the second Hankel determinant for the class $\mathcal{U}$ and the general class $\mathcal{S}$

TL;DR

This paper establishes upper bounds for specific Hankel determinants within the classes of univalent functions and the subclass al, providing new insights into their coefficient constraints.

Contribution

It determines the upper bounds of the second Hankel determinants for the classes al and al, extending the understanding of these determinants for special subclasses.

Findings

Upper bounds for $H_2(3)(f)$ in al

Upper bounds for $H_2(4)(f)$ in al

Results applicable to the general class al

Abstract

In this paper we determine the upper bounds of the Hankel determinants of special type $H_{2}(3)(f)$ and $H_{2}(4)(f)$ for the class of univalent functions and for the class $\mathcal{U}$ defined by \[ \mathcal{U}=\left\{ f\in\mathcal{A} : \left|\left[\frac{z}{f(z)}\right]^2 f'(z)-1 \right|<1,\, z\in{\mathbb D} \right\}, \] where $\mathcal{A}$ is the class of functions analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+\cdots$.