Generalization of Mrs. Gerber's Lemma

Mrs. Gerber’s Lemma (MGL) hinges on the convexity of H(p ∗ H(u)), where H(u) is the binary entropy function. In this work, we prove thatH(p∗f(u)) is convex in u for every p ∈ [0, 1] provided H(f(u)) is convex in u, where f(u) : (a, b) → [0, 1 2 ]. Moreover, our result subsumes MGL and simplifies the original proof. We show that the generalized MGL can be applied in binary broadcast channel to simplify some discussion.