Coefficient Bounds and Stability for an Arctangent–Superexponential Convolution Class

This paper presents a new convolution framework for normalized analytic functions in the open unit disk. The framework uses an arctangent and superexponential coefficient kernel, with each part serving a different purpose. The arctangent factor controls the early Taylor coefficients, while the cubic superexponential factor quickly reduces the higher-order terms. Using the logarithmic-derivative starlikeness condition on the result of this operator, we define an operator-defined quartic polynomial-domain family: the transformed function is starlike, while the starlikeness of the original function is not assumed. This approach offers a parameter-sensitive extension of Ma–Minda-type starlike classes and creates a link between coefficient-weighted operators and geometric subordination. The principal findings include initial coefficient bounds, a Fekete–Szegö bound with explicit parameter dependence, and a sufficient—not necessary—coefficient criterion obtained from a disk contained in the quartic polynomial domain. A radius-dependent refinement, a general invariance theorem, and a general perturbation theorem then provide a concise framework for coefficient transformations and stability; symmetry projections, diagonal damping, residue-class multipliers, and blockwise perturbations are treated as direct consequences.