HTTP adaptive streaming (HAS) has emerged as the main technology for video streaming applications. Multiple HAS video clients sharing the same wireless channel may experience different video qualities, as well as, different play-out buffer levels, as a result of both different video content complexities and different channel conditions. This causes unfairness in the end-user quality of experience (QoE). In this paper, we propose a quality-fair adaptive streaming solution with fair buffer (QFASFB) to deliver fair video quality and to achieve asymptotically fair play-out buffer levels among HAS clients competing for the same wireless resources in an LTE cell. In the QFAS-FB framework the share of radio resources is optimized according to video content characteristics, play-out buffer levels and channel conditions. The proposed solution is compared with other state-of-the-art strategies and the numerical results show that it significantly improves the quality fairness among heterogeneous HAS users, it reduces the video quality variations, and improves the fairness among the user's play-out buffers.