Poincaré inequality 3/2 on the Hamming cube

Abstract

For any $n\ge 1$, and any $f:\{-1,1{\}}^n\to \mathbb{R}$, we have

where $z^{3/2}$ for $z=x+iy$ is taken with principal branch, and $\mathcal{R}$ denotes the real part. We show an application of this inequality: it sharpens a well-known inequality of Beckner.