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.
翻译：我们强化了韩氏傅里叶熵-影响不等式 $$ H[\widehat{f}] \leq C_{1}I(f) + C_{2}\sum_{i\in [n]}I_{i}(f)\ln\frac{1}{I_{i}(f)} $$，该不等式最初针对取值为$\{-1,1\}$的布尔函数，在$C_{1}=3+2\ln 2$和$C_{2}=1$的条件下得证。通过简短的信息论证明，我们表明该不等式实际上对所有具有单位$L^{2}$范数的实值布尔函数均成立，且具有最优常数$C_{1}=C_{2}=1$，从而确立了该不等式作为香农熵与影响的基本结构性质。
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