Green cognitive radio (CR) plays an important role in offering secondary users (SUs) with more spectrum with smaller energy expenditure. However, the energy efficiency (EE) issues associated with green CR for fading channels have not been fully studied. In this paper, we investigate the average EE maximization problem for spectrum-sharing CR in fading channels. Unlike previous studies that considered either the peak or the average transmission power constraints, herein, we considered both of these constraints. Our aim is to maximize the average EE of SU by optimizing the transmission power under the joint peak and average transmit power constraints, the rate constraint of SU and the quality of service (QoS) constraint of primary user (PU). Specifically, the QoS for PU is guaranteed based on either the average interference power constraint or the PU outage constraint. To address the non-convex optimization problem, an iterative optimal power allocation algorithm that can tackle the problem efficiently is proposed. The optimal transmission powers are identified under both of perfect and imperfect channel side information (CSI). Simulations show that our proposed scheme can achieve higher EE over the existing scheme and the EE achieved under perfect CSI is better than that under imperfect CSI.