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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
DOI: https://doi.org/10.1090/S0025-5718-1985-0790653-0
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$.


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

  • [1] P. J. Bushell & A.H. Craven, "On Shapiro's cyclic inequality," Proc. Roy. Soc. Edinburgh, Sect. A, v. 26, 1975/76, pp. 333-338. MR 0480320 (58:493)
  • [2] D. E. Daykin, "Inequalities for functions of a cyclic nature," J. London Math. Soc. (2), v. 3, 1971, pp. 453-462. MR 0284394 (44:1622a)
  • [3] P. H. Diananda, "On a cyclic sum," Proc. Glasgow Math. Assoc., v. 6, 1963, pp. 11-13. MR 0150084 (27:87)
  • [4] D. Ž. Djoković, "Sur une inégalité," Proc. Glasgow Math. Assoc., v. 6, 1963, pp. 1-10. MR 0150083 (27:86)
  • [5] C. V. Durell, "Query," Math. Gaz., v. 40, 1956, p. 266. MR 0075977 (17:827h)
  • [6] E. K. Godunova & V. I. Levin, "A cyclic sum with 12 terms," Math. Notes, v. 19, 1976, pp. 510-517. MR 0424578 (54:12537)
  • [7] J. C. Lagartas, "The van der Waerden conjecture: Two Soviet solutions," Notices Amer. Math. Soc., v. 29, 1982, pp. 130-133.
  • [8] M. A. Malcolm, "A note on a conjecture of L. J. Mordell," Math. Comp., v. 25, 1971, pp. 375-377. MR 0284395 (44:1622b)
  • [9] D. S. Mitrinović, Analytic Inequalities, Springer-Verlag, New York, 1970, pp. 132ff. MR 0274686 (43:448)
  • [10] L. J. Mordell, "On the inequality $ \Sigma _{r = 1}^n {x_r}/({x_{r + 1}} + {x_{r + 2}}) \geqslant \frac{1}{2}n$ and some others," Abh. Math. Sem. Univ. Hamburg, v. 22, 1958, pp. 229-240. MR 0096915 (20:3397)
  • [11] P. Nowosad, "Isoperimetric eigenvalue problems in algebra," Comm. Pure Appl. Math., v. 21, 1968, pp. 401-465. MR 0238087 (38:6363)
  • [12] 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)
  • [13] H. S. Shapiro, "Problem 4603," Amer. Math. Monthly, v. 61, 1954, p. 571. MR 1528827
  • [14] A. Zulauf, "Note on a conjecture of L. J. Mordell," Abh. Math. Sem. Univ. Hamburg, v. 22, 1958, pp. 240-241. MR 0124261 (23:A1575)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1985-0790653-0
Keywords: Cyclic inequality, cyclic sum, minimization
Article copyright: © Copyright 1985 American Mathematical Society

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