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Anantharam, Jog and Nair recently unified the Shannon-Stam inequality and the entropic form of the Brascamp-Lieb inequalities under a common inequality. They left open the problems of extremizability and characterization of extremizers. Both questions are resolved in the present paper. PreliminariesWe begin by briefly fixing notation and definitions that will be needed throughout. A Euclidean space E is a finite-dimensional Hilbert space over the real field, equipped with Lebesgue measure. For a probability measure µ on E, absolutely continuous with respect to Lebesgue measure, and a random vector X ∼ µ, we define the Shannon entropy h(X) ≡ h(µ) := − E