Let me take a stab at this. Let's define the function F as

We will show that the minumum of this is zero. Taking partial derivatives and setting to zero gives

By substracting in pairs we obtain

from which we deduce that

Substituting into any of the partial derivatives gives

giving rise to three cases:

The first gives whereas the remaining two gives . Thus, the minimum of F is 0 giving

for which the inequality follows.