A class of Lyapunov functions for discrete-time Lurie systems with monotonic non-linearities is proposed. The Lyapunov functions are composed of quadratic terms on the states and of the system's non-linearities as well as Lurie-Postnikov type integral terms. Crucially, positive definiteness of the matrix in the generalised quadratic form and positivity of the scaling terms of the Lurie-Postnikov integrals are relaxed in the stability conditions. Furthermore, they are used for regional stability analysis and performance assessment. Numerical examples show that the proposed Lyapunov function structure matches or outperforms existing ones for these systems.