Abstract We discuss some frequency-domain criteria for the exponential stability of nonlinear feedback systems based on Popov’s hyperstability theorem. The main results are new bounds on convergence rates for perturbations of the damped harmonic oscillator, including the Hessian-damped case, when the perturbations satisfy certain sectoriality or convexity hypotheses. We also include a simplified proof of Popov’s theorem in its general form, which has the virtue that it often applies at the margins of stability when other criteria typically degenerate.