A variance form of Han's bound on the Fourier entropy

Han proved that every Boolean function f satisfies \[ H(f)≤ 3/ln 2\,I(f)+∑_k Inf_k(f)log_24/Inf_k(f), \] where H is the Fourier entropy and I the total influence. We show that the logarithm may be taken relative to the variance: \[ H(f)≤ (2+3/ln 2)I(f)+∑_k Inf_k(f)log_2(1+Var(f)/Inf_k(f)). \] The bound implies Han's up to an additive $I(f)$ and is smaller by a factor of order k for the AND of k variables. It follows that the Fourier entropy-influence conjecture holds, with constant 2+3/ln2+log₂(1+1/ε), for every function whose influences are all at least (f), and that H(f)≤ I(f)\,(3+3/ln 2+H(π)), where πₖ=Infₖ(f)/I(f) is the influence distribution. The proof combines the chain rule for the Fourier entropy along a decision tree with a one-coordinate estimate, and an adaptive version holds along any decision tree.