Minkowski's triangle inequality for $0I have a trouble in the following theorem introduced "Minkowski's inequality for $0<p<1$":

Let $0<p<1$ and let $x,y\ge0$. Then, $|x^{p}-y^{p}|\le|x-y|^{p}$.

I have proved the case for $p=\frac{1}{2}$.
How do I prove it in the general case ?
Give some advice! Thank you!

A concave function $f:[0,\infty)\to\Bbb R$ such that $f(0)\ge0$ must be subadditive: if by any chance $f(a+b)>f(a)+f(b)$, then $\frac{f(a+b)-f(a)}{b}>\frac{f(b)-f(0)}{b}$, against the hypothesis of concavity.
So, for $p\in(0,1)$ and $x,y\ge 0$, we have that $$(\max\{x,y\}-\min\{x,y\})^p+(\min\{x,y\})^p\ge (\max\{x,y\})^p\\(\max\{x,y\}-\min\{x,y\})^p\ge(\max\{x,y\})^p-(\min\{x,y\})^p\\ \lvert x-y\rvert^p\ge(\max\{x,y\})^p-(\min\{x,y\})^p$$
And, since $x^p$ is an increasing function, the RHS is $\lvert x^p-y^p\rvert$.
