In some remote areas under extremely scarce computation and communication resources, unmanned aerial vehicles (UAVs) assisted wireless communications are quite attractive for the more widely communication coverage and powerful computation capacity in contrast to user mobile devices (MDs). In addition to providing relay communication capabilities, the UAV can not only perform user tasks locally, but also offload them to base station or cloudlet for computing, provided that the user tasks require more powerful computation capacities. Considering the fact that the mobility trajectory of UAV could exert a negative impact on mobile offloading, we investigate the mobile offloading problem with a comparative consideration of mobility trajectory, communication, and computation resource allocation, in which the UAV-assisted mobile offloading and trajectory optimization problem (UOTO) is formulated with the objective of maximizing minimum user utility. Unfortunately, the UOTO provided is proven to be non-convex and there is no effective way to solve it. To overcome the difficulties, we develop a near-optimal algorithm by integrating the nonlinear fractional programming (NFP) and successive convex approximation (SCA). The extensive experiments are conducted to illustrate the performance of the proposed scheme.