An operator-coefficients free Poincaré inequality

We prove the following operator-coefficients free Poincaré inequality: $$\mathrm{\mid }f(X)-E[f(X)]{\mathrm{\mid }}_2\le 2\mathrm{\mid }X{\mathrm{\mid }}_2\mathrm{\parallel }{\widehat{\mathrm{\partial }}}_{X:B}[f(X)]{\mathrm{\parallel }}_{\pi },f(X)\in \mathrm{d}\mathrm{o}\mathrm{m}({\widehat{\mathrm{\partial }}}_{X:B}),$$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\pi }$ is the projective tensor norm.