On Third-Order Determinant Bounds for the class $\mathcal{S}^*_{B}$

TL;DR

This paper establishes sharp bounds for third-order Hankel, Toeplitz, and Hermitian-Toeplitz determinants of a specific class of starlike functions, expanding understanding of their coefficient constraints and extremal properties.

Contribution

It provides the first sharp bounds for these determinants within the class al S^*_B, using coefficient inequalities and extremal function construction.

Findings

Sharp bounds for third-order determinants are derived.

Extremal functions are constructed to verify sharpness.

Results enhance understanding of coefficient constraints in al S^*_B.

Abstract

This paper deals with sharp bounds for the third-order Hankel, Toeplitz and Hermitian-Toeplitz determinant of functions belonging to the class $\mathcal{S}^*_{B}$ of starlike functions associated with a balloon-shaped domain, given by \[ \mathcal{S}^{\ast}_{B}= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1}{1-\log (1+z)} :=B(z), \quad z \in \mathbb{D} \right\}. \] By applying coefficient inequalities and properties of these functions, we obtain sharp bounds for these determinants. The sharpness of the results is verified by constructing suitable extremal functions.