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