The KKL inequality and Rademacher type 2

We show that a vector-valued Kahn–Kalai–Linial inequality holds in every Banach space of Rademacher type 2. We also show that for any nondecreasing function $h\ge 0$ with $0<{\int }_1^{\mathrm{\infty }}\frac{h(t)}{t^2}\mathrm{d}\mathrm{t}<\mathrm{\infty }$ we have the inequality \begin{align*} |f - $\mathbb{E}$f|2 \leq 12 , T{2}(X) \left(\int_{1}^{\infty}\frac{h(t)}{t^{2}} $\mathrm{d}\mathrm{t}$ \right)^{1/2} , \left(\sum_{j=1}^n \frac{|D_j f|^{2}_2}{h\left( \log \frac{|D_j f|_2}{|D_j f|1} \right)}\right)^{1/2} \end{align*} for all $f :{-1,1}^{n} \to X$andall$n\geq 1$,where$X$isanormedspaceand$T{2}(X)$ is the associated type 2 constant.