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.