Strengthening Han's Fourier Entropy-Influence Inequality via an Information-Theoretic Proof

We strengthen Han’s Fourier entropy-influence inequality $$H[\widehat{f}]\le C_{1}I(f)+C_2\sum _{i\in [n]}{I}_i(f)\ln \frac{1}{I_i(f)}$$ originally proved for $\{-1,1\}$-valued Boolean functions with $C_1=3+2\ln 2$ and $C_2=1$. We show, by a short information-theoretic proof, that it in fact holds with sharp constants $C_1=C_2=1$ for all real-valued Boolean functions of unit $L^2$-norm, thereby establishing the inequality as an elementary structural property of Shannon entropy and influence.