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

2020/04/19 by Andrzej Czarnecki, Czarnecki, Andrzej, Gabriel Kiciński +1
Mathematics · Social Sciences · #Mathematical Inequalities and Applications #Names, Identity, and Discrimination Research #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2004.08882

Abstract

We prove that the cyclic inequality\n\∑\_i=1ⁱ⁼ⁿ\( fracx_ix_i+1\)^k\≥\∑\_i=1ⁱ⁼ⁿ fracx_ix_\σ(i)\nholds for k in a specific range dependant on the permutation \σ. We\nalso show that the same is not true for the Sahpiro-type generalizations.\n

Related

• Cyclic inequalities involving Cater cyclic function
• Beyond Shapiro's problem: from cyclic sums to "graphic'' sums
• On a Cyclic Inequality Related to Chebyshev Polynomials
• Lower bound for cyclic sums with one-sided maximal averages in denominators
• Weighted Sequences in Finite Cyclic Groups