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In this paper, we consider the spectrum sharing planning problem for a full-duplex unmanned aerial vehicle (UAV) relaying systems with underlaid device-to-device (D2D) communications, where a mobile UAV employed as a full-duplex relay assists the communication link between separated nodes without direct link. Our design aims to maximize the sum throughput under the transmit power budget, while guaranteeing the coexistence with terrestrial D2D pairs, satisfying the information causality and UAV's trajectory constraints. First, the transmit power planning with a given trajectory is investigated, where a successive convex algorithm is developed by leveraging the D.C. (difference of two convex) programming. Then, we propose a two-step trajectory design method for the given transmit power since the constraints of D2D pairs result in a non-convex feasible set. Furthermore, an efficient spectrum sharing method …