Poincaré inequality 3/2 on the Hamming cube (Q2177512)

Summary: For any \(n \geq 1\), and any \(f \colon \{-1,1\}^n \to \mathbb{R} \), we have \[\mathfrak R \mathbb{E} (f + i |\nabla f|)^{3/2} \leq \mathfrak R (\mathbb{E}f)^{3/2},\] where \(z^{3/2}\) for \(z=x+iy\) is taken with principal branch, and \(\mathfrak R\) denotes the real part. We show an application of this inequality: it sharpens a well-known inequality of \textit{W. Beckner} [Proc. Am. Math. Soc. 105, No. 2, 397--400 (1989; Zbl 0677.42020)].