We investigate the use of parametrized families of information-theoretic measures to generalize the loss functions of generative adversarial networks (GANs) with the objective of improving performance. A new generator loss function, called least $k$th-order GAN (L$k$GAN), is first introduced, generalizing the least squares GANs (LSGANs) by using a $k$th order absolute error distortion measure with $k \geq 1$ (which recovers the LSGAN loss function when $k=2$). It is shown that minimizing this generalized loss function under an (unconstrained) optimal discriminator is equivalent to minimizing the $k$th-order Pearson-Vajda divergence. Another novel GAN generator loss function is next proposed in terms of Rényi cross-entropy functionals with order $α>0$, $α\neq 1$. It is demonstrated that this Rényi-centric generalized loss function, which provably reduces to the original GAN loss function as $α\to1$, preserves the equilibrium point satisfied by the original GAN based on the Jensen-Rényi divergence, a natural extension of the Jensen-Shannon divergence. Experimental results indicate that the proposed loss functions, applied to the MNIST and CelebA datasets, under both DCGAN and StyleGAN architectures, confer performance benefits by virtue of the extra degrees of freedom provided by the parameters $k$ and $α$, respectively. More specifically, experiments show improvements with regard to the quality of the generated images as measured by the Fréchet Inception Distance (FID) score and training stability. While it was applied to GANs in this study, the proposed approach is generic and can be used in other applications of information theory to deep learning, e.g., the issues of fairness or privacy in artificial intelligence.