Sharp Coefficient Bounds for a Disk-Starlike Preimage Class Defined by an Algebraically Damped Hadamard Operator

Coefficient operators provide a natural way to reshape an analytic function before testing its geometry. This paper develops a two-parameter algebraically damped Hadamard-product operator and investigates the functions whose transformed images satisfy a disk-starlikeness condition. The construction retains a visible contribution from the first coefficients while allowing the higher-order tail to be attenuated at an adjustable algebraic rate. The geometric conclusion concerns the transformed function; it does not automatically imply that the original function is starlike or univalent. Sharp initial-coefficient estimates, a corrected Fekete–Szegö inequality, and corresponding extremal functions are obtained. The multiplier is compared quantitatively with several established coefficient operators, and the effects of its parameters on damping and coefficient flexibility are examined. A weighted-coefficient criterion yields transparent perturbation and convex-combination consequences, while explicit examples illustrate both membership and failure of the criterion. The practical profiles included in the paper are conceptual visualizations rather than empirically validated models.