I have noticed in the literature that Young's inequality $x^2+y^2 \geq 2xy$ is mentioned considering non-negative $x$ and $y$. But considering various cases, I have found that the inequality holds for negative non zero $x$ and $y$. Does the Young's inequality holds for negative $x$ or $y$ in general?
Case:1 $x=-1$ and $y=1$. $\implies1+1\geq-1$ (Inequality holds).
Case:2 $x=1$ and $y=-1$. $\implies 1+1\geq-1$ (Inequality holds).
Case:3 $x=-1$ and $y=-1$. $\implies 1+1 \geq 1$ (Inequality holds)