The widespread application of mobile electronic devices has triggered a boom in energy consumption, especially in user equipment (UE). In this paper, we investigate the energy-efficiency (EE) of a UE experiencing the worst channel conditions, which is termed worst-EE. Due to the limited battery of the mobile equipment, worst-EE is a suitable metric for EE fairness optimization in the uplink transmissions of orthogonal frequency division multiple access (OFDMA) networks. More specifically, we determine the optimal power and sub-carrier allocation to maximize the worst-EE with respect to UEs' transmit power, sub-carriers and statistical quality-of-service (QoS). In order to maximize the worst-EE, we formulate a max-min power and sub-carrier allocation problem, which involves nonconvex fractional mixed integer nonlinear programming, i.e., NP-hard to solve. To solve the problem, we first relax the allocation of sub-carriers, formulate the upper bound problem on the original one and prove the quasi-concave property of objective function. With the aid of the Powell-Hestenes-Rockfellar (PHR) approach, we propose a fairness EE sub-carrier and power allocation algorithm. Finally, simulation results demonstrate the advantages of the proposed algorithm.