Poincaré inequality 3/2 on the Hamming cube

Abstract: For any $n \geq 1$, and any $f :{-1,1}^{n} \to \mathbb{R}$ we have $ \Re\, \mathbb{E}\, (f + i\, |\nabla f|)^{3/2} \leq \Re\, (\mathbb{E}f)^{3/2}, $ where $z^{3/2}$ for $z=x+iy$ is taken with principal branch and $\Re$ denotes the real part.