Simple inequality in C*-algebras

Theorem 1.5 of this 1987 paper by J. Phillips says that if $f:[0,infty)to [0,infty)$ is a continuous operator monotone function and $a$ and $b$ are positive operators on a Hilbert space, then $|f(a)-f(b)|leq f(|a-b|)-f(0)$. I think that the proof is nice. Corollary 1.6 says that $|a^{1/n}-b^{1/n}|leq|a-b|^{1/n}$, $ngeq1.$ Of course your inequality follows from taking $a=x^2$, $b=y^2$, and $n=2$.

Apparently Kittaneh and Kosaki have a similar approach in "Inequalities for the Schatten p-norm. V." Publ. Res. Inst. Math. Sci. 23 (1987), no. 2, 433--443 (MR link). I haven't read any of this article.

Perhaps I should add the following for a more general audience. A continuous function $f:[0,infty)to [0,infty)$ is operator monotone if whenever $x$ and $y$ are positive operators such that $y-x$ is positive, it follows that $f(y)-f(x)$ is positive. The functions $tmapsto t^alpha$ are operator monotone for $0<alphaleq1$ (but not for $alpha>1$).