On a cyclic inequality with exponents and permutations, and its Shapiro-type analogues
Tom 127 / 2021
Annales Polonici Mathematici 127 (2021), 201-209
MSC: 26D15, 26D20.
DOI: 10.4064/ap210119-6-9
Opublikowany online: 9 November 2021
Streszczenie
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.
