Sharp bounds of Hankel determinants of second and third order for inverse functions of certain class of univalent functions

TL;DR

This paper establishes the precise upper bounds for the second and third order Hankel determinants of inverse functions of a specific class of univalent functions, advancing the understanding of their coefficient bounds.

Contribution

It provides the first sharp bounds for these Hankel determinants for inverse functions within the class ${ m U}(\lambda)$, a significant extension in geometric function theory.

Findings

Sharp bounds for second order Hankel determinants

Sharp bounds for third order Hankel determinants

Enhanced understanding of inverse functions in ${ m U}(\lambda)$

Abstract

Let ${\mathcal A}$ be the class of functions that are analytic in the unit disc ${\mathbb D}$, normalized such that $f(z)=z+\sum_{n=2}^\infty a_nz^n$, and let class ${\mathcal U}(\lambda)$, $0<\lambda\le1$, consists of functions $f\in{\mathcal A}$, such that \[ \left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < \lambda\quad (z\in {\mathbb D}). \] In this paper we determine the sharp upper bounds for the Hankel determinants of second and third order for the inverse functions of functions from the class ${\mathcal U}(\lambda)$.