Strengthening Han's Fourier Entropy-Influence Inequality via an Information-Theoretic Proof
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915661703282688 |
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| author | Li, Peijie Han, Guangyue |
| author_facet | Li, Peijie Han, Guangyue |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_03117 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strengthening Han's Fourier Entropy-Influence Inequality via an Information-Theoretic Proof Li, Peijie Han, Guangyue Information Theory 94D10 (Primary) 94A15 (Secondary) 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. |
| title | Strengthening Han's Fourier Entropy-Influence Inequality via an Information-Theoretic Proof |
| topic | Information Theory 94D10 (Primary) 94A15 (Secondary) |
| url | https://arxiv.org/abs/2512.03117 |