Sharpness Estimation of Hankel Determinants and Logarithmic Coefficients for a Family of Analytic Functions Related to a Lung-Shaped Domain

This investigation introduces a novel family of univalent analytic functions subordinate to lung-shaped domains within the open unit disk. Through rigorous application of subordination theory and systematic analysis, we establish coefficient bounds for the initial five coefficients, derive estimates for Hankel determinants of orders two and three, determine bounds for the first four logarithmic coefficients, and derive the bounds of some Zalcman functionals. The lung-shaped domain is characterized by the subordination condition involving a secant-based function, which maps the unit disk onto a geometrically distinctive region exhibiting bilateral symmetry. All obtained bounds are demonstrated to be sharp through the construction of specific extreme functions.