Within the UAV network, the UAV first receives signals from multiple remote mobile devices (MDs) and then amplifies and forwards the transmitted signals to the base station (BS) with different amplification coefficients to form a UAV relay multiuser network. In this paper, we propose a new method to solve the problem of maximizing the throughput of the relay network. The proposed problem is decoupled into two subproblems, UAV deployment design and amplification coefficient optimization, to be solved iteratively, respectively. We solve the UAV deployment problem by adjusting its trajectory with a gradient descent-based method and solve the amplification coefficient subproblem with a convex optimization-based method iteratively. Simulation shows that the proposed UAV deployment and amplification coefficient of the UAV optimization design algorithm significantly improves the sum-rate compared with existing fixed relay and equal power allocation schemes. Finally, we discuss future potential performance enhancing methods including multiple UAV cooperation, massive multi-input multioutput (MIMO) communications, and nonorthogonal multiple access (NOMA) communications.