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\ddot{o}lderonselfadjointelementsofnoncommutative$L_p$-spacesfor$0<p\leq\infty$withvaluesin$L_{p/\theta}$. This extends an inequality of Birman, Koplienko and Solomjak also obtained by Ando.