In this paper, we adopt an energy-efficiency (EE) metric, named worst-EE, that is suitable for EE fairness optimization in the uplink transmission of amplify-and-forward (AF) cooperative orthogonal frequency division multiple access (OFDMA) networks. More specifically, we assign subcarriers and allocate powers for mobile and relay stations in order to maximize the worst-EE, i.e., to maximize the EE of the mobile station (MS) with the lowest EE value, subject to MSs transmit power, relay station (RS) transmit power, and MSs quality-of-service (QoS) constraints. The formulated primal max-min optimization problem is nonconvex fractional mixed integer nonlinear program, i.e., NP-hard to solve. We provide a novel optimization framework that studies the structure of the primal problem and prove that the dual min-max optimization problem attains the same optimal solution of the primal problem. Additionally, we propose a modified Dinkelbach algorithm, named dual Dinkelbach, to achieve the optimal solution of the dual problem in a polynomial time complexity. We further exploit the structure of the obtained optimal solution and develop a low complexity suboptimal heuristic. Numerical results show the effectiveness of the proposed algorithm to improve the network performance in terms of fairness between MSs, worst-EE, and average network transmission rate when compared to traditional schemes that maximize the EE of the whole network. Presented results also show that the suboptimal heuristic balances the achieved performance and the computational complexity.