Per-flow unfairness of TCP throughput in the IEEE 802.11 wireless LAN (WLAN) environment has been reported in past literature. A number of researchers have proposed various methods for alleviating the unfairness; most require modification of MAC protocols or queue management mechanisms in access points. However, the MAC protocols of access points are generally implemented at hardware level, so changing these protocols is costly. As the first contribution of this paper, we propose a transport-layer solution for alleviating unfairness among TCP flows, requiring a small modification to TCP congestion control mechanisms only on WLAN stations. In the past literature on fairness issues in the Internet flows, the performance of the proposed solutions for alleviating the unfairness has been evaluated separately from the network bandwidth utilization, meaning that they did not consider the trade-off relationships between fairness and bandwidth utilization. Therefore, as the second contribution of this paper, we introduce a novel performance metric for evaluating trade-off relationships between per-flow fairness and bandwidth utilization at the network bottleneck. We confirm the fundamental characteristics of the proposed method through simulation experiments and evaluate the performance of the proposed method through experiments in real WLAN environments. We show that the proposed method can achieve better a trade-off between fairness and bandwidth utilization, regardless of vendor implementations of wireless access points and wireless interface cards.