We propose Distributional Concavity (DC) regularization for Generative Adversarial Networks (GANs), a functional gradient-based method that promotes the entropy of the generator distribution and works against mode collapse. Our DC regularization is an easy-to-implement method that can be used in combination with the current state of the art methods like Spectral Normalization and Wasserstein GAN with gradient penalty to further improve the performance. We will not only show that our DC regularization can achieve highly competitive results on ILSVRC2012 and CIFAR datasets in terms of Inception score and Fr\'echet inception distance, but also provide a mathematical guarantee that our method can always increase the entropy of the generator distribution. We will also show an intimate theoretical connection between our method and the theory of optimal transport.