The Fourier entropy-influence inequality for symmetric Boolean functions

We prove that every symmetric Boolean function f satisfies H[f] ≤ Csym I[f] with Csym = 2.692 + 2 + 1/ln 2 < 6.1347, improving the constant 12.04 of O'Donnell, Wright and Zhou. The Fourier entropy of a symmetric function splits into the entropy of its level weights and a within-level term. The first is at most 2.692 I[f] for every Boolean function, by a sharpened spectral-level estimate whose last step is a one-variable inequality checked by interval arithmetic. For the second we prove that the derivative of a symmetric function has noise stability at most e−vI0(v) times its squared norm, where v is the variance of the noise, and evaluate the resulting entropy integral through a Frullani identity and two inequalities for modified Bessel functions. The AND and OR functions show that the optimal constant for symmetric functions is at least 4. We also prove that the entropy cost of restricting one coordinate of a symmetric function is asymptotically at most 4.637 times its influence, so the restriction step that would give constant 4.8723 can fail in only finitely many dimensions.