On the Third Hankel Determinant for Inverse Coefficients of Starlike Functions: A Bernstein Polynomial Approach

TL;DR

This paper establishes a sharp upper bound for the third Hankel determinant of inverse coefficients of starlike univalent functions using Bernstein polynomial methods.

Contribution

It introduces a Bernstein polynomial approach to derive the first sharp upper bound for the third Hankel determinant in this context.

Findings

Derived a sharp upper bound for the third Hankel determinant.

Applied Bernstein polynomial method to inverse coefficients of starlike functions.

Enhanced understanding of coefficient bounds for univalent functions.

Abstract

Let $\mathcal{A}$ denote the class of normalized analytic functions $f$ in the open unit disk defined as $ \mathbb{D}:=\{z\in\mathbb{C}:|z|<1\} $ with $f(0)=0$ and $f'(0)=1$. A function $f\in\mathcal{A}$ is said to be starlike if $f(\mathbb{D})$ is starlike domain. By using the Bernstein polynomial method to obtain the required maximum estimate, we establish sharp upper bound for the third Hankel determinant corresponding to the inverse coefficients of starlike univalent ({\it i.e.}, one-to-one) functions in the unit disk $\mathbb{D}$.