How to prove inequality similar to that of Minkowski

Question

I am struggling with this problem. How to prove the below inequality

$|x+y|^r$ $\leq$ $|x|^r+|y|^r$ for $0<r\leq1$.

A little guidance please.

Answer 1

If $x = 0$ or $y = 0$ or $x+y=0$, then it is trivially true. Assume $x,y,x+y \neq 0$ then we prove:

$1 \leq \left|\dfrac{x}{x+y}\right|^r+\left|\dfrac{y}{x+y}\right|^r = a^r + b^r, a+b \geq 1 $.

Consider $f(r) = a^r + b^r \to f'(r) = \ln r(a^r+b^r) < 0 \to f(r) \geq f(1) =a+b \geq 1$.