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

Volume 127 / 2021

Andrzej Czarnecki, Gabriel Kiciński Annales Polonici Mathematici 127 (2021), 201-209 MSC: 26D15, 26D20. DOI: 10.4064/ap210119-6-9 Published online: 9 November 2021

Abstract

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.

Authors

  • Andrzej CzarneckiFaculty of Mathematics and Computer Science
    Jagiellonian University
    Łojasiewicza 6
    30-848 Kraków, Poland
    e-mail
  • Gabriel KicińskiFaculty of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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