For some $w$,$x$,$y$,$z$ it holds that $w+x>y+z$. Does that mean that $\sqrt w+\sqrt x> \sqrt y+\sqrt z$ and the other way around? For some $w$,$x$,$y$,$z$ it holds that 
$$w+x>y+z. \quad \quad (1)$$Does that mean that 
$$\sqrt w+\sqrt x> \sqrt y+\sqrt z \quad \quad (2)$$ and the other way around, so if (2) holds then (1) too?
Thanks!

No.  Counter-examples found by computer search:
$$1 \implies 2 \text{ counter-example } (w, x, y, z) = (1, 6, 3, 3)$$
$$2 \implies 1 \text{ counter-example } (w, x, y, z) = (2, 2, 1, 3)$$
