Question

Let P(x) be a polynomial with non-negative coefficients. Prove that if the inequality P(1/x)P(x)≥1 holds for x=1, then this inequality holds for all positive x.

Answer 1

I feel like a proof of this is just saying the obvious over and over. There's nothing insightful or illuminating about this--it just seems so strikedly true. I've wrote up three different proofs and they just looked like rubbish because it felt so obvious.

@KingGeorge and @nbouscal, can either of you give this some sort of insight I am missing? I would like some higher level reasoning than appealing to its naively true-esque look (which I seem to be doing, unfortunately).

Answer 2

Sorry, guys. I see now that there are alternate solutions here http://openstudy.com/users/badreferences#/updates/4f3af03ee4b0fc0c1a0ed1ac . If you two would still like to give a proof, feel free.