I know that it holds for non-negatives but consider the following:
The inequality holds for non-negatives but it holds for real numbers as well iff since:
(Generally Note that we can prove the same way that )
Considering that the inequality holds for all real numbers by rearrangement Inequality, I suppose that something similar for the inequality should exist. But unfortunately is not the condition we are looking for...
I don't really know if that helps but