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A simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) enhanced unnamed aerial vehicle (UAV)-enabled multi-user mobile edge computing (MEC) scheme is proposed in this paper. Different from the existing MEC works, the proposed scheme allows bi-directional offloading where users can simultaneously offload their computing tasks to the MEC servers situated at the ground base station (BS) and aerial UAV with the assistance of the STARRIS. Specifically, we formulate an optimization problem aiming at maximizing the energy efficiency of the system while ensuring the quality of service (QoS) constraint by jointly optimizing the resource allocation, user scheduling, passive beamforming of the STAR-RIS, and the UAV trajectory. An iterative algorithm designed with the Dinkelbach's algorithm and the successive convex approximation (SCA) is proposed to effectively handle the formulated non-convex optimization problem. Simulation results indicate that the proposed STAR-RIS enhanced UAV-enabled MEC scheme possesses significant advantages in enhancing the system energy efficiency over other baseline schemes including the conventional RIS-aided scheme.