Remote Access Mathematics of Computation
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] D. E. DAYKIN, "Inequalities for functions of cyclic nature," J. London Math. Soc. (2), v. 3, 1971, pp. 453-462. MR 0284394 (44:1622a)
  • [2] P. H. DIANANDA, "On a cyclic sum," Proc. Glasgow Math. Assoc., v. 6, 1963, pp. 11-13. MR 0150084 (27:87)
  • [3] P. H. DIANANDA, "A cyclic inequality and an extension of it. II," Proc. Edinburgh Math. Soc. (2), v. 13, 1962/1963, pp. 143-152. MR 0148831 (26:6335)
  • [4] D. Ž. DJOKOVIĆ, "Sur une inégalité," Proc. Glasgow Math. Assoc., v. 6, 1963, pp. 1-10. MR 0150083 (27:86)
  • [5] D. S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, New York, 1970, p. 132 ff. MR 0274686 (43:448)
  • [6] P. NOWOSAD, "Isoperimetric eigenvalue problems in algebra," Comm. Pure Appl. Math., v. 21, 1968, pp. 401-465. MR 0238087 (38:6363)
  • [7] R. A. RANKIN, "An inequality," Math. Gaz., v. 42, 1958, pp. 39-40.
  • [8] J. L. SEARCY & B. A. TROESCH, "A cyclic inequality and a related eigenvalue problem," Pacific J. Math., v. 81, 1979, pp. 217-226. MR 543745 (80f:15019)
  • [9] H. S. SHARIRO, "Problem 4603," Amer. Math. Monthly, v. 61, 1954, p. 571. MR 1528827

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10E20

Retrieve articles in all journals with MSC: 10E20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1980-0551296-2
Keywords: Cyclic inequality, shooting method, minimization
Article copyright: © Copyright 1980 American Mathematical Society