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!

If $\,x=y=0\,$ the inequality holds trivially, otherwise assume WLOG that $\,x \ge y\,$ and let $\,t = y/x \in [0,1]\,$. After dividing by $\,x^p \gt 0\,$ the inequality then becomes $\,1-t^p \le (1-t)^p\,$.
Since $\,t^p \ge t\,$ and $\,(1-t)^p \ge 1-t\,$ for $\,0 \lt p \lt 1\,$ and $\,0 \le t, 1-t \le 1\,$, it follows that: $$\,1-t^{p} \;\le\; 1-t \;\le\; (1-t)^{p}\,$$
