Many conditions have been found for the absolute stability of discrete-time Lur'e systems in the literature. It is advantageous to find convex searches via LMIs where possible. In this technical note, we construct two less conservative LMI conditions for discrete-time systems with slope-restricted nonlinearities. The first condition is derived via Lyapunov theory while the second is derived via the theory of integral quadratic constraints (IQCs) and noncausal Zames-Falb multipliers. Both conditions are related to the Jury-Lee criterion most appropriate for systems with such nonlinearities, and the second generalizes it. Numerical examples demonstrate a significant reduction in conservatism over competing approaches.