Motivated by the need to ensure timely delivery of information (e.g., status updates) in the Internet-of-things (IoT) paradigm, this paper investigates the role of an Unmanned aerial vehicle (UAV) as a mobile relay to minimize the average Peak Age-of-information (PAoI) for a source-destination pair. For this setup, we formulate an optimization problem to jointly optimize the UAV's flight trajectory as well as energy and service time allocations for packet transmissions. In order to solve this non-convex problem, we propose an efficient iterative algorithm and establish its convergence analytically. Closed-form solutions for some sub-problems are also provided. One of the sub-problems we solve in this procedure is to jointly optimize the energy and service time allocations for a given trajectory of the UAV. This problem is of interest on its own right because in some cases we may not be able to alter the UAV's trajectory based on the locations of the IoT devices (especially when its primary mission is something else). Our numerical results quantify the gains that can be achieved by additionally optimizing the UAV's trajectory.