Given the imperative of 6G networks' ubiquitous connectivity, along with the inherent mobility and cost-effectiveness of unmanned aerial vehicles (UAVs), UAVs play a critical role within 6G wireless networks. Despite advancements in enhancing the UAV-enabled communication systems' throughput in existing studies, there remains a notable gap in addressing issues concerning user fairness and quality-of-service (QoS) provisioning and lacks an effective scheme to depict the trade-off between system throughput and user fairness. To solve the above challenges, in this paper we introduce a novel fairness control scheme for UAV-enabled wireless communication systems based on a new weighted function. First, we propose a throughput combining model based on a new weighted function with fairness considering. Second, we formulate the optimization problem to maximize the weighted sum of all users' throughput. Third, we decompose the optimization problem and propose an efficient iterative algorithm to solve it. Finally, simulation results are provided to demonstrate the considerable potential of our proposed scheme in fairness and QoS provisioning.