In this paper, a new subcarrier allocation algorithm is proposed for an uplink multiuser OFDM relay network. To achieve fairness for a source and relay, we consider the minimum rate requirement for a source and the power constraint on a relay, respectively. An optimization problem with fairness constraints is formulated and the subcarrier allocation criterion is obtained by using convex optimization. In the proposed algorithm, if all sources meet the minimum rate requirement, a subcarrier is allocated to a source-relay pair which has the largest instantaneous rate to maximize the sum of the achievable rate of a source. Otherwise, a subcarrier is allocated to a source-relay pair which has the smallest achievable rate to improve fairness for a source. Simulation results show that the proposed algorithm improves fairness and spectral efficiency for a source.