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The validity of Shapiro's cyclic inequality


Author: B. A. Troesch
Journal: Math. Comp. 53 (1989), 657-664
MSC: Primary 26D15
DOI: https://doi.org/10.1090/S0025-5718-1989-0983563-0
MathSciNet review: 983563
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Abstract: A cyclic sum $ {S_N}({\mathbf{x}}) = \sum {{x_i}/({x_{i + 1}} + {x_{i + 2}})} $ is formed with N components of a vector x, where in the sum $ {x_{N + 1}} = {x_1}$, $ {x_{N + 2}} = {x_2}$, and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors x for which $ {S_N}({\mathbf{x}}) < N/2$ if $ N \geq 14$ and even, and if $ N \geq 24$. It has been proved that the inequality $ {S_N}({\mathbf{x}}) \geq N/2$ holds for $ N \leq 13$. Although it has been conjectured repeatedly that the inequality also holds for odd N between 15 and 23, this has apparently never been proved. Here we will confirm that the inequality indeed holds for all odd $ N \leq 23$. This settles the question for all N.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1989-0983563-0
Keywords: Cyclic inequality, cyclic sum, minimization
Article copyright: © Copyright 1989 American Mathematical Society

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