In this paper, we investigate a mobile edge computing (MEC) architecture with the assistance of an unmanned aerial vehicle (UAV). The UAV acts as a computing server to help the user equipment (UEs) compute their tasks as well as a relay to further offload the UEs' tasks to the access point (AP) for computing. The total energy consumption of the UAV and UEs is minimized by jointly optimizing the task allocation, the bandwidth allocation and the UAV's trajectory, subject to the task constraints, the information-causality constraints, the bandwidth allocation constraints, and the UAV's trajectory constraints. The formulated optimization problem is nonconvex, and we propose an alternating algorithm to optimize the parameters iteratively. The effectiveness of the algorithm is verified by the simulation results, where great performance gain is achieved in comparison with some practical baselines, especially in handling the computation-intensive and latency-critical tasks.