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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The shooting method applied to a cyclic inequality


Author: B. A. Troesch
Journal: Math. Comp. 34 (1980), 175-184
MSC: Primary 10E20
MathSciNet review: 551296
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Abstract: It is known that the cyclic sum

$\displaystyle {S_n}({\mathbf{x}}) = \sum\limits_{i = 1}^{i = n} {{x_i}/({x_{i + 1}} + {x_{i + 2}})} $

where $ {x_{n + 1}} = {x_1}$, $ {x_{n + 2}} = {x_2}$, $ {x_i} \geqslant 0$, $ ({x_{i + 1}} + {x_{i + 2}}) > 0$, can be made smaller than $ n/2$ for $ n \geqslant 24$. The value of $ \lambda (n) = \lim \inf \;(n \to \infty )\;{S_n}/n$ is investigated by the shooting method for two-point boundary value problems, and the analytical result $ \lim (n \to \infty )\;\lambda (n) \leqslant 0.49457$ is proved. The inherent difficulty in a straight-forward minimization of $ {S_n}({\mathbf{x}})$ is mentioned.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1980-0551296-2
PII: S 0025-5718(1980)0551296-2
Keywords: Cyclic inequality, shooting method, minimization
Article copyright: © Copyright 1980 American Mathematical Society