A new extremal entropy inequality with applications

Liu et al. proved an extremal entropy inequality using a vector generalization of the Costa entropy-power inequality (EPI). The generalized Costa EPI was proved, in turn, using a perturbation approach via a fundamental relationship between the derivative of mutual information and the minimum mean-square error (MMSE) estimate in linear vector Gaussian channels. In this paper, we consider two new variations of the (Liu et al.) extremal entropy inequality. Instead of employing perturbation approaches, we employ a new method recently introduced by Geng and Nair, which was used to resolve the capacity region of the Gaussian MIMO broadcast channel (BC) with common and private messages. As an application, we use one of the extremal entropy inequalities to prove the capacity region of a class of reversely degraded Gaussian MIMO BC with three users and three-degraded message sets.