Fractional powers on noncommutative $L_p$ for $p<1$

We prove that the homogeneous functional calculus associated to $x\mapsto |x|^\theta$ or $x\mapsto {\rm sgn}\, (x) |x|^{\theta}$ for $0<\theta<1$ is $\theta$-H\"older on selfadjoint elements of noncommutative $L_p$-spaces for $0<p\leq\infty$ with values in $L_{p/\theta}$. This extends an inequality of Birman, Koplienko and Solomjak also obtained by Ando.