In this paper, we consider the synthesis of fixed order controllers for nonlinear systems with sector bounded nonlinearities. We construct an inner and outer approximation of the set of absolutely stabilizing linear controllers by casting the closed loop system as a Lure-Postnikov system. The inner approximation is based on the well-known sufficient conditions that require strict positive realness (SPR) of open loop transfer function (possibly with some multipliers) and a characterization of SPR transfer functions that require a family of complex polynomials to be Hurwitz. The outer approximation is based on the condition that the open loop transfer function must have infinite gain margin, which translates to a family of real polynomials being Hurwitz. We illustrate the proposed methodology through the construction of an inner and outer approximation of absolutely stabilizing controllers for a mechanical system.