The shooting method applied to a cyclic inequality
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- by B. A. Troesch PDF
- Math. Comp. 34 (1980), 175-184 Request permission
Abstract:
It is known that the cyclic sum \[ {S_n}({\mathbf {x}}) = \sum \limits _{i = 1}^{i = n} {{x_i}/({x_{i + 1}} + {x_{i + 2}})} \] 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.References
- D. E. Daykin, Inequalities for functions of a cyclic nature, J. London Math. Soc. (2) 3 (1971), 453–462. MR 284394, DOI 10.1112/jlms/s2-3.3.453
- P. H. Diananda, On a cyclic sum, Proc. Glasgow Math. Assoc. 6 (1963), 11–13 (1963). MR 150084
- P. H. Diananda, A cyclic inequality and an extension of it. II, Proc. Edinburgh Math. Soc. (2) 13 (1962/63), 143–152. MR 148831, DOI 10.1017/S0013091500014711
- D. Ž. Djoković, Sur une inégalité, Proc. Glasgow Math. Assoc. 6 (1963), 1–10 (1963) (French). MR 150083
- D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686
- Pedro Nowosad, Isoperimetric eigenvalue problems in algebras, Comm. Pure Appl. Math. 21 (1968), 401–465. MR 238087, DOI 10.1002/cpa.3160210502 R. A. RANKIN, "An inequality," Math. Gaz., v. 42, 1958, pp. 39-40.
- J. L. Searcy and B. A. Troesch, A cyclic inequality and a related eigenvalue problem, Pacific J. Math. 81 (1979), no. 1, 217–226. MR 543745
- H. S. Shapiro, Richard Bellman, D. J. Newman, W. E. Weissblum, H. R. Smith, and H. S. M. Coxeter, Advanced Problems and Solutions: Problems for Solution: 4603-4607, Amer. Math. Monthly 61 (1954), no. 8, 571–572. MR 1528827, DOI 10.2307/2307617
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Math. Comp. 34 (1980), 175-184
- MSC: Primary 10E20
- DOI: https://doi.org/10.1090/S0025-5718-1980-0551296-2
- MathSciNet review: 551296