*Problem:*

Let be positive real numbers with sum equal to . Find the minimum value of the expression

.

*Solution:*

From the definition of , we have that

.

Solving the 1st Inequality we get that

.

Solving now the 2nd Inequality we have that

.

The last relation reduces to the

.

So, we see now that for .

Summing up these Inequalities we acquire

or, due to the hypothesis we now have that

.

If we solve this last Inequality it gives us the result , which is also and the value we are searching, *Q.E.D.*

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