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This paper investigates the robust downlink beamforming designs based on max-min fairness and base station (BS) power consumption constraint for device-to-device (D2D) communications underlaying cellular network, under the assumption of imperfect channel state information (CSI) at the BS. Our objective is to maximize the minimum signal-to-interference-plus-noise ratio (SINR) or non-outage probability of users while guaranteeing that the consumed power at the BS is less than a threshold. In particular, three max-min fairness scenarios are considered. In the first scenario, the worst-case SINR of cellular users (CUs) is maximized where the D2D SINR is guaranteed to be above a specified predetermined threshold. In the second scenario, we extend the first scenario and maximized D2D SINR while maximizing CUs' SINR. In these two scenarios, it is assumed that the errors are upper bounded in their Frobenius norms. In the third scenario, the minimum non-outage probability of users is maximized, while a probabilistic model is considered for the uncertainty of channel covariance matrices. Although such optimization problems are not convex, the semidefinite relaxation (SDR) approach is used to obtain the optimal beamforming matrix, which always complies the rank-one constraint. Simulation results show that significant improvement in terms of minimum SINR of CUs, the minimum non-outage probability of CUs, and the probability of feasibility and the fairness index can be achieved with our proposed algorithms in comparison with other beamforming schemes. K E Y W O R D Sdevice-to-device communications, imperfect channel state information, max-min fairness, robust beamforming