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Adaptive bitrate video streaming (ABR) has become a critical technique for mobile video streaming to cope with time-varying network condi-tions and different user preferences. However, there are still many problems in achieving high-quality ABR video streaming over cellular networks. Mo-bile Edge Computing (MEC) is a promising paradigm to overcome the above problems by providing video transcoding capability and caching the ABR video streaming within the radio access network (RAN). In this paper, we propose a flexible transcoding strategy to provide viewers with low-latency video streaming services in the MEC networks under the limited stor-age, computing, and spectrum resources. According to the information collected from users, the MEC server acts as a controlling component to adjust the transcod-ing strategy flexibly based on optimizing the video caching placement strategy. Specifically, we cache the proper bitrate version of the video segments at the edge servers and select the appropriate bitrate ver-sion of the video segments to perform transcoding un-der jointly considering access control, resource alloca-tion, and user preferences. We formulate this problem as a nonconvex optimization and mixed combinato-rial problem. Moreover, the simulation results indicate that our proposed algorithm can ensure a low-latency viewing experience for users.