Sharp Estimates of Hankel Determinants for certain classes of convex univalent functions

TL;DR

This paper derives precise bounds for the second and third Hankel determinants of certain convex univalent functions defined via subordination, identifying extremal functions for these bounds.

Contribution

It provides sharp estimates of Hankel determinants for a class of convex functions defined by a specific subordination condition, including extremal functions.

Findings

Sharp bounds for second and third Hankel determinants are established.

Extremal functions achieving these bounds are explicitly characterized.

The results extend understanding of geometric properties of convex univalent functions.

Abstract

Let $\mathcal{A}$ denote the class of analytic functions $f$ such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ We examine the properties of the class $\mathcal{C}(\varphi)$ defined as $\mathcal{C}(\varphi) := \left\{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec \varphi(z):=1+z+ m/n\, \, z^2, \text{ with } 2m \le n,\text{ for } m, n \in \mathbb{N} \right\},$ and compute the sharp second and third Hankel determinants for the functions in $\mathcal{C}(\varphi)$. Furthermore, we determine the extremal functions for the sharp estimates of the Hankel determinants.