Quadratic Onset of Radial Negative Depth for High-Gain Carathéodory Polynomials

How much positivity must be lost when a high-gain Carathéodory polynomial is continued past its maximal radius of positivity? Let have nonnegative real part in the unit disk and satisfy for fixed . Classical Fejér–Riesz and Kac–Murdock–Szegő theory gives a microscopic positivity threshold at radius . At radius we study the smallest possible normalized negative depth over this class. We prove that this class minimum converges, as , to a band-limited extremal value under inverse-Poisson evolution; that exactly for ; and that the post-threshold onset is quadratic, as . The lower bound follows from Fejér-window localization of the classical nonnegative band-limited extremal inequality. The upper bound is realized by a diffuse perturbation of height and width ; its normalized first-order perturbations lose tightness, allowing an mass correction while preserving inner positivity.