In this paper we make explicit connections between classical absolute stability theory and modern mixed-mu analysis and synthesis. Specifically, using the parameter-dependent Lyapunov function framework of Haddad and Bernstein and the frequency dependent off-axis circle interpretation of How and Hall, we extend previous work on absolute stability theory for monotonic and odd monotonic nonlinearities to provide tight approximations for constant real parameter uncertainty. An immediate application of this framework is the generalization and reformulation of mixed-mu analysis and synthesis in terms of Lyapunov functions and Riccati equations. This observation is exploited to provide robust, reduced-order controller synthesis while avoiding the standard D, N - K iteration and curve-fitting procedures.