This letter investigates fairness-aware task data allocation and trajectory optimization in unmanned aerial vehicles (UAV)-assisted mobile edge computing (MEC) systems, where a fixed-wing UAV is used as a flying computing server to receive task data of mobile terminals (MTs). Under the fairness consideration, we aim to minimize the maximum energy consumption among all MTs. Despite the non-convexity of the original formulated joint optimization problem, we transform the problem into two convex sub-problems by introducing auxiliary variables, and solve them jointly by proposing an iterative algorithm. Simulation results show that the proposed algorithm can effectively reduce the maximum energy consumption among all MTs.