show this sum $\sum(x_{i})^{x_{i+1}}\le n^{(n-1)/n}$

Question

conjectures let $x_{i}(i=1,2,\cdots,n)$,and such $x_{1}+x_{2}+\cdots+x_{n}=1$,show that $$(x_{1})^{x_{2}}+(x_{2})^{x_{3}}+\cdots+(x_{n})^{x_{1}}\le n^{\frac{n-1}{n}}$$

It seem use Jensen inequality:because when $x_{i}=\dfrac{1}{n}$ this inequality is indentity

Answer 1

I think your inequality is wrong, try $n=4, x_1=1^-, x_2=x_3=x_4=0^+$. $$x_1^{x_2}+x_2^{x_3}+x_3^{x_4}+x_4^{x_1}=1+1+1+0=3>2\sqrt2=n^{{n-1}/n}$$