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Mathematics of Computation

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On Shapiro's cyclic inequality for $ N=13$

Author: B. A. Troesch
Journal: Math. Comp. 45 (1985), 199-207
MSC: Primary 26D15; Secondary 05A20
MathSciNet review: 790653
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Abstract: A cyclic sum $ S({\mathbf{x}}) = \Sigma \;{x_i}/({x_{i + 1}} + {x_{i + 2}})$ is formed with the N components of a vector x, where $ {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({\mathbf{x}}) < N/2$ if $ N \geqslant 14$ and even, and if $ N \geqslant 25$. It has been proved that the inequality $ S({\mathbf{x}}) \geqslant N/2$ holds for $ N \leqslant 12$. Although it has been conjectured repeatedly that the inequality also holds for odd N between 13 and 23. this has apparently not yet been proved. Here we will confirm that the inequality indeed holds for $ N = 13$.

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Keywords: Cyclic inequality, cyclic sum, minimization
Article copyright: © Copyright 1985 American Mathematical Society