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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Keywords:
Cyclic inequality,
cyclic sum,
minimization

Article copyright:
© Copyright 1989
American Mathematical Society