In this note, the problem of state feedback H-infinity control for a class of nonlinear systems is considered. The class under study is a generalization of the well-known Lur'e systems. The H-infinity problem is addressed via a class of storage functions of the Lur'e-Postnikov type whose integral term is parameterized by a nonlinear scalar function. The related H-infinity controllers consist of a linear term, which is designed for the underlying linearized system, plus a nonlinear term which depends on the nonlinear function. A simple geometrical criterion is provided for the characterization of the set of controllers which ensure a given level of L-2-performance globally. Some guidelines for an effective design of the controller within this set are discussed via two examples.