Sharp Bound of the Third-Order Hankel Determinant for a Symmetric Domain

This paper presents a new approach to the third-order Hankel determinant problem for the family of starlike functions associated with a symmetric H-domain. Motivated by a recently proposed sharp bound, we develop an analytical method that substantially improves the best known estimate. The obtained upper bound is reduced from 0.104485 to 0.062581, which is remarkably close to the conjectured sharp constant 1/16=0.0625. This significant improvement provides strong evidence supporting the validity of the conjectured bound while introducing a systematic technique for deriving refined coefficient estimates. Furthermore, the proposed approach is sufficiently flexible to be extended to related extremal problems involving Hankel determinants and other subclasses of analytic functions in geometric function theory.