Let are non-negative numbers, no two of them are zero. Prove that
Let us multiply each fraction cyclic by respectively. Then we acquire that .
Now from Cauchy-Schwartz inequality we get
So, we only need to prove that
Let us denote by the , that is .
We have that: .
On the other hand:
But . So,
So, we finally have that
Multiplying by we aqcuire , .
From we deduce that
But from Schur’s inequality we know that
, and .
Adding up these inequalities we kill our inequality, Q.E.D.