The second and third Hankel determinants for certain convex subclass of functions

TL;DR

This paper computes the exact second and third Hankel determinants for a specific subclass of convex functions defined via a subordination condition involving a quadratic function.

Contribution

It introduces a new subclass of convex functions and derives the sharp bounds for their second and third Hankel determinants.

Findings

Computed sharp bounds for second Hankel determinant.

Computed sharp bounds for third Hankel determinant.

Established the extremal functions achieving these 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 the present paper, we consider $\mathcal{C}(\varphi) := \left\{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec \varphi(z):=(1+z/2)^2 \right\}$, as subclass of convex functions and compute the sharp second and third Hankel determinants for functions in $\mathcal{C}(\varphi)$.