On a cyclic inequality with exponents and permutations, and its Shapiro-type analogues

We prove that the cyclic inequality $\sum _{i=1}^{n}\bigl (\frac {x_i}{x_{i+1}}\bigr )^k\geq \sum _{i=1}^{n}\frac {x_i}{x_{\sigma (i)}}$ holds for all positive $x_i$’s if and only if $k$ is in a specific range depending on the permutation $\sigma $, related to band permutations. We also show that the same is not true for Shapiro-type generalizations, proving in the process some analogous inequalities with exponents.