*Problem:*

Let be non-negative real numbers. Prove that

.

*1st solution:*

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:

.

*2nd solution:*

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.

*3nd solution:*

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

, *Q.E.D.*

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