Friedgut–Kalai Fourier entropy-influence conjecture

Let $f:\{-1,1\}^n\to\{-1,1\}$ be a Boolean function with normalized Fourier coefficients, Fourier entropy $H[f]$, and total influence $I[f]$.

Friedgut–Kalai Fourier entropy-influence conjecture. There exists a universal constant $c$ such that, for every such $f$,

$$H[f]\leq c I[f].$$

This is described as a major open problem in the analysis of Boolean functions. The conjecture concerns a dimension-independent comparison between Fourier entropy and total influence.