A technique is proposed to address the robust stability of a certain class of nonlinear systems. It is based on conic confinement of the nonlinear part of the dynamics. The tight (nonconservative) bounding cones are characterized, and an algorithm is provided. After that, the nonlinearity is modelled as a complex uncertainty lying inside a small-gain cone. Stability under additional arbitrary uncertainties in the linear part, given as frequencial QFT templates, is addressed by robust QFT techniques, deriving conditions in the form of frequencial boundaries for the controller.