I am interested in this question. Squaring out some terms, I get the following inequality:
Definitely not what we are looking for!!
Notice that the inequality we need to prove contains a term in . Thus we have to try somehow or other to bring that term into our working.
Now, since we are working with non-negative numbers, we can multiply all the inequalities together. Let’s try that.
The LHS is equal to (expand them out and check). The RHS is equal to . Hence
Which doesn’t bring us any closer to a solution, but at least we now have the term to work with.
I'm sure this is supposed to be a slick application of one of those Jensen-type inequalities that they drill math olympiad candidates with. Here's a much more pedestrian approach (aesthetically unappealing because it breaks the symmetry).
If and then and . Thus . In particular, . Also, . Then
. . . . . .