For , prove
Chebyshev's inequality we'll have:
but by AM-GM we have:
imagine how many inequalities you can create using only Chebyshev's inequality + AM-GM inequality! i guess i'm back to high school again!
I know that this inequality can be shown by special inequalities, the first one dropped in my mind was Muirhead's inequality (again because of the symmetry), which you may find in Muirhead's inequality - Wikipedia, the free encyclopedia.
If there is any other solutions, I would like to see.
anyway, yes, Muirhead's inequality will also solve the problem because if you multiply both sides of the inequality by you'll get:
which is true because majorizes