TITLE: Please explain inequality $|x^{p}-y^{p}| \leq |x-y|^p$
QUESTION [17 upvotes]: Suppose $x \geq 0$, $y \geq 0$ and $0<p<1$. Why is the following inequality true?
$|x^{p}-y^{p}| \leq |x-y|^p$

REPLY [15 votes]: This follows from the more general inequality
\begin{equation}(1)\qquad\qquad|x+y|^p\le |x|^p+|y|^p\qquad\qquad\text{(for $x,y\in\mathbb{C}$ and $0\lt p\le1$)}\end{equation}
Indeed, if we replace $x$ by $x-y$ in (1) we get 
$$|x|^p\le |x-y|^p+|y|^p$$
which imply
$$|x|^p-|y|^p\le |x-y|^p$$
To prove (1), first note
$$|x+y|^p\le(|x|+|y|)^p$$
Hence it is sufficient to prove (1) for $x,y\ge0$ in which case we may apply Sivaram's argument in the previous answer.