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

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 $f^{\prime }(0)=1$. 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 $\mathbb{D}$.