Sharp extremal bounds and membership-profile stability in fuzzy geometric subordination of analytic functions

Using nested level domains and their normalized conformal maps, we create fuzzy geometric constraints. The initial coefficients and the Fekete–Szegő functional in the corresponding starlike and convex classes are sharply bounded for right-half-plane dominants by the first two Taylor coefficients. We derive explicit sharp growth, derivative-distortion, and covering estimates for half-plane levels, with extremal functions achieving the bounds. Additionally, we provide quantitative perturbation estimates for geometric envelopes and coefficient constants; the latter exhibit uniform convergence on compact subdisks. The findings show that while crisp membership recovers the classical classes, the membership profile regulates both extremal constants and image geometry. In this study, when target boundaries or tolerance levels are uncertain, normalized analytic maps of the unit disk are taken into consideration as geometric models. The radius r tracks the location where growth is measured, the profile beta establishes the order of the half-plane constraint, and the dimensionless membership parameter alpha records the confidence level. Epsilon measures profile perturbation, lambda chooses a coefficient functional, and the Taylor coefficients characterize the local map shape. Level selection, geometric classification, local and global estimates, and sensitivity analysis are all supported by these parameters, respectively.