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

We prove that the cyclic inequality ${\sum }_{i=1}^{i=n}{\left(\frac{x_i}{x_{i+1}}\right)}^k\ge {\sum }_{i=1}^{i=n}\frac{x_i}{x_{\sigma (i)}}$ holds for $k$ in a specific range dependant on the permutation $\sigma$. We also show that the same is not true for the Sahpiro-type generalizations.