The inequality $\|x^p-y^p\|\leq\|x-y\|^p$ holds even when $x$ and $y$ are positive operators on Hilbert space. This follows from a more general result proved as Theorem 1.5 in the paper [1] by J. Phillips, which gives the inequality $\|f(x)-f(y)\|\leq f(\|x-y\|)$ whenever $f$ is an operator monotone function on $[0,\infty)$ such that $f(0)=0$. The fact that $t\mapsto t^p$ is operator monotone if $0\lt p\leq 1$ is proven as Proposition 1.3.8 in Pedersen's C*-algebras book. Your inequality is the $1$-dimensional case. [1]: https://dspace.library.uvic.ca:8443/dspace/handle/1828/1506 – 2010-11-20