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

TL;DR

This paper provides a simplified, information-theoretic proof that strengthens Han's Fourier entropy-influence inequality, establishing sharp constants and broadening its applicability to real-valued Boolean functions.

Contribution

It offers a concise proof demonstrating the inequality with optimal constants for all real-valued Boolean functions, enhancing understanding of the relationship between entropy and influence.

Findings

Proves the inequality with sharp constants C1=C2=1

Extends the inequality to all real-valued Boolean functions

Provides an elementary proof based on information theory

Abstract

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.