an inequality with a cyclic shift of exponents

Asked Aug 19 '21 at 11:06

Active Aug 23 '21 at 07:17

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Prove that $$ x_1^{x_1}+x_2^{x_2}+\ldots+x_n^{x_n}\ge x_1^{x_2}+x_2^{x_3}+\ldots+x_n^{x_1} $$ for any $x_1,\dots,x_n>0$.

I found this problem in a contest-preparing ebook. There was a solution I wasn't able to understand, it required some results from multivariable calculus. I was trying to solve it with less advanced stuff, but with no sensible result to show.

edited Aug 19 '21 at 11:28

Jakobian

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asked Aug 19 '21 at 11:06

user957042

1

I think we can just use greedy theorem. Raise the greatest number to the greatest power and so on. – acat3 Aug 19 '21 at 11:29
I have a proof for $n=2$ and $n=3.$ :( – Michael Rozenberg Aug 19 '21 at 12:22
@MichaelRozenberg Is this for $n = 2$?: https://math.stackexchange.com/questions/4226202/prove-that-aabb-ge-abba1-if-a-b-0 – River Li Aug 19 '21 at 14:27
@River Li Yes, but this proof is wrong. – Michael Rozenberg Aug 19 '21 at 14:29
@MichaelRozenberg Whatever, $n = 2$ is easy. – River Li Aug 19 '21 at 14:32
@River Li Not so easy, but it's smooth enough. A proof for $n=3$ is not so easy already. For $n=4$ I have no a proof. – Michael Rozenberg Aug 19 '21 at 14:34
https://math.stackexchange.com/questions/1325361/prove-the-inequality-x-1y-1x-2y-2-cdotsx-ny-n-geq-x-1y-pi1?noredirect=1 – user265131 Aug 19 '21 at 21:51

an inequality with a cyclic shift of exponents

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