Necessary and sufficient conditions for some inequalities to hold
For (1):
$(\implies)$ Set $a=f$ and $b=h$. Since $c\geq f+h$, $f\leq f$, and $h\leq h$, the claim holds.
$(\impliedby)$ Since $f\leq a$ and $h\leq b$ we have, $f+h\leq a+b$. Hence, $c\geq a+b\geq f+h$.
For (2), it is proven similarly.