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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The validity of Shapiro’s cyclic inequality

Author: B. A. Troesch
Journal: Math. Comp. 53 (1989), 657-664
MSC: Primary 26D15
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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Keywords: Cyclic inequality, cyclic sum, minimization
Article copyright: © Copyright 1989 American Mathematical Society