A closed-form expression for the Sharma-Mittal entropy of exponential families

The Sharma-Mittal entropies generalize the celebrated Shannon, R'enyi and Tsallis entropies. We report a closed-form formula for the Sharma-Mittal entropies and relative entropies for arbitrary exponential family distributions. We instantiate explicitly the formula for the case of the multivariate Gaussian distributions and discuss on its estimation.