Let be non-negative real numbers. Prove that
The inequality is homogeneous, so, if we normalize it we get that .
Doing some manipulation in the left hand side we acquire:
Let us denote by the . Then we have that:
But , so:
Thus the previous inequality can be rewritten as .
Now, from Schur’s inequality we know that
So, we come to the conclusion:
Doing all the manipulations in the left and in the right hand side we only need to prove that
But the last one holds because it is equivalent to:
whose first terms hold from Muirhead’s inequality and the last one from Schur’s inequality.
Let . Then the inequality takes the following form:
Doing the manipulations in the left hand side we get that
Thus we obtain that
From here we obtain that
Let us denote by the and the similarly.
Without loss of generality assume that . From here we have that
For the end of our proof we only need to show that ,
which reduces to