Sorry the title is a bit vague. Let A be a C*-algebra, and let x and y be positive elements in A. Is it true that $$ \|x-y\|^2 \leq \|x^2-y^2\|? $$ Well, yes. But the proof I have is a bit of a hack, so I wonder if anyone has a "nice" proof, or a reference?

Aside: if $A=C_0(X)$ then this reduces to the inequality $(a-b)^2 \leq |a^2-b^2|$ for non-negative real numbers a and b.

**Update:** Jonas points me to http://www.springerlink.com/content/j4756m418220644r/ where Kittaneh has a proof pretty similar to what I had in mind (unpack the proof of Theorem 1). I guess I was interested in whether this sort of thing was standard (if I looked in the right textbook) or if it was a bit of a curiosity. I think the latter seems more likely...