Analytic Speed Limits and Extremal Rigidity for Strip-Holomorphic Monotone Profiles

We study real-valued transition profiles on the real axis that admit holomorphic extension to a horizontal strip in the complex plane. The functions considered have a continuously differentiable and nondecreasing real trace and are normalized to take values strictly between zero and one. We assume that the associated conformal transform obtained by rescaling and shifting the profile extends holomorphically to the strip and maps it into the unit disk. Under these conditions strip analyticity imposes a sharp pointwise bound on the rate of change along the real axis. The bound depends only on the width of the analytic strip and is optimal. We further prove a rigidity result: if the bound is attained at any real point then the profile is uniquely determined up to translation and coincides with the logistic transition. The argument is purely analytic and follows from the Schwarz–Pick contraction principle applied to the strip geometry. No classification of non-saturating profiles is attempted.