The second and third Hankel determinants for starlike MA--Minda subclass associated to quadratic polynomials

TL;DR

This paper investigates the properties and bounds of second and third Hankel determinants for a specific starlike subclass of analytic functions related to quadratic polynomials, establishing sharp bounds and extremal functions.

Contribution

It introduces a new subclass of starlike functions defined via quadratic polynomial subordination and computes sharp bounds for their second and third Hankel determinants.

Findings

Computed explicit bounds for second and third Hankel determinants.

Identified extremal functions that achieve the bounds.

Established sharpness of the bounds.

Abstract

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}$. In this paper, we discuss the properties of a starlike subclass and compute its second and third Hankel determinants; where the class is defined as $\mathcal{S}^*(\varphi):=\{f\in\mathcal{A}:{zf'(z)}/{f(z)}\prec \varphi(z):=1+z+{m}/{n}\,\, z^2,\text{ such that } 2m \le n, \text{ where } m,n\in\mathbb{N}\}.$ Furthermore, we show that the bounds are sharp by determining the extremal functions for the Hankel determinants.