The Tsypkin and Jury-Lee criteria are commonly used to analyse the absolute stability of discrete-time Lur'e systems with slope-restricted nonlinearities. In this paper, we construct a corresponding linear matrix inequality (LMI) condition for the Jury-Lee criterion most appropriate for monotonic, slope-restricted nonlinearities. The corresponding Lur'e-Lyapunov function is also constructed and, via the Lyapunov method, the conditions on the aforementioned criterion are relaxed. The result is explicitly compared with improved LMI-based criteria in the literature. The resulting multiplier from the criterion is also shown to satisfy the conditions of the Zames-Falb multipliers in discrete-time. This indirectly provides a convex search over a subset of the discrete-time Zames-Falb multipliers. Some numerical examples for SISO and MIMO cases are provided to compare the performance of the criteria with existing results, and we demonstrate that the result in this paper provides significant improvement