In this paper, we focus on an unmanned aerial vehicle (UAV) assisted communication network, where a UAV is dispatched to transmit date to multi-ground-devices (GDs). We provide a fairness-aware design on UAV trajectory and user assignment, intending to maximize the minimal throughput, while a practical Rician fading model is taken into account. To address the non-convex formulated problem with infinite variables, we first apply the successive-hover-and-fly (SHF) structure, helping to reformulate the problem a finite variables problem. Although the new problem is still highly non-convex, we apply the successive convex approximation (SCA) technique where a set of tight concave functions are constructed, approximating the reformulated problem into a convex form, based on which, an iterative algorithm is enabled to solve the reformulated problem to obtain an efficient solution. Finally, numerical results are provided to validate the high performance of our proposed algorithm in throughput and algorithm computing time cost.