We strengthen Han's Fourier entropy-influence inequality $$ H[\widehat{ f}] \leq C_{ 1}I(f) + C_{ 2}\sum_{ i\in [n]}I_{ i}(f)\ln\frac{ 1}{ I_{ i}(f)} $$ 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.
Categories: cs.IT math.IT