Research on undersampled magnetic resonance image (MRI) reconstruction can increase the speed of MRI imaging and reduce patient suffering. In this paper, an undersampled MRI reconstruction method based on Generative Adversarial Networks with the Self-Attention mechanism and the Relative Average discriminator (SARA-GAN) is proposed. In our SARA-GAN, the relative average discriminator theory is applied to make full use of the prior knowledge, in which half of the input data of the discriminator is true and half is fake. At the same time, a self-attention mechanism is incorporated into the high-layer of the generator to build long-range dependence of the image, which can overcome the problem of limited convolution kernel size. Besides, spectral normalization is employed to stabilize the training process. Compared with three widely used GAN-based MRI reconstruction methods, i.e., DAGAN, DAWGAN, and DAWGAN-GP, the proposed method can obtain a higher peak signal-to-noise ratio (PSNR) and structural similarity index measure(SSIM), and the details of the reconstructed image are more abundant and more realistic for further clinical scrutinization and diagnostic tasks.