The second and third Hankel determinants for certain classes of functions

TL;DR

This paper computes the sharp second and third Hankel determinants for a specific subclass of starlike functions defined by a subordination condition involving a quadratic function.

Contribution

It provides explicit calculations of the second and third Hankel determinants and identifies extremal functions within the subclass of starlike functions defined by a particular subordination.

Findings

Sharp second and third Hankel determinants are obtained.

Extremal functions for coefficient bounds are determined.

Results contribute to understanding the coefficient structure of this function class.

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 consider $\mathcal{S}^*(\varphi) := \left\{ f \in \mathcal{A} : zf'(z)/f(z) \prec \varphi(z):=(1+z/2)^2 \right\}$, a subclass of starlike functions and we compute the sharp second and third Hankel determinants for the functions in $\mathcal{S}^*(\varphi)$. Furthermore, we determine the extremal functions for the coefficient bounds of the functions belonging to $\mathcal{S}^*(\varphi)$.