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I'm sorry for the silly question, but I have a doubt. Given two positive real numbers $x,y$ and taking $0<r<1$, is it true that
$$ (x+y)^r \le x^r+y^r? $$
In all the examples I considered, this turns out to be true. Moreover, to try to give a proof, does it suffices to prove that the function $f(z)=(1+z)^r-1-z^r$ is non-positive when $z \in (0,1]$ (i.e. taking $x>y$ and dividing by $x$?