By the symmetry we may assume, WLOG, that
Then
So, by Chebyshev's inequality we have:
And now, by Nesbitt's Inequality we have:
Thus
By the symmetry we may assume, WLOG, that
Then
So, by Chebyshev's inequality we have:
And now, by Nesbitt's Inequality we have:
Thus
Lemma. Suppose we have finite sequences: of positive real numbers
Then we have: ( this is a particular case of Hölder's inequality, I'm proving it because I do not know whether you are familiar with it or not)
Proof
By AM-GM inequality: ( for )
Thus:
( set ... above)
Now sum the inequalities:
Thus: and the rest follows easily.
Now let's go to your inequality.
By the lemma we have:
Thus:
Now note that:
Thus we have:
But the following inequality holds: (*)
Therefore:
To prove (*) note that:
Similarly: and sum those inequalities.