Given that , , for the numbers , show that .

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- Jan 1st 2008, 10:22 AMjames_bondInequality
Given that , , for the numbers , show that .

- Jan 2nd 2008, 06:00 AMcolby2152
I am interested in this question. Squaring out some terms, I get the following inequality:

Definitely not what we are looking for!!:confused: - Jan 2nd 2008, 07:07 AMIsomorphism
- Jan 2nd 2008, 12:43 PMJaneBennet
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. :rolleyes: - Jan 4th 2008, 11:16 AMOpalg
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

. . . . . . - Jan 5th 2008, 07:03 PMThePerfectHacker
- Jan 5th 2008, 08:31 PMIsomorphism