## The validity of Shapiro’s cyclic inequality

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- by B. A. Troesch PDF
- Math. Comp.
**53**(1989), 657-664 Request permission

## 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*.

## References

- P. H. Diananda,
*On a cyclic sum*, Proc. Glasgow Math. Assoc.**6**(1963), 11–13 (1963). MR**150084**, DOI 10.1017/S2040618500034626 - 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**, DOI 10.1017/S2040618500034614 - V. G. Drinfel′d,
*A certain cyclic inequality*, Mat. Zametki**9**(1971), 113–119 (Russian). MR**280660** - J. Lambek and L. Moser,
*Rational analogues of the logarithm function*, Math. Gaz.**40**(1956), 5–7. MR**75977**, DOI 10.2307/3610258 - E. K. Godunova and V. I. Levin,
*A cyclic sum with twelve terms*, Mat. Zametki**19**(1976), no. 6, 873–885 (Russian). MR**424578**
J. C. Lagarias, "The van der Waerden conjecture: Two Soviet solutions," - 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**, DOI 10.1007/978-3-642-99970-3 - 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,
*A cyclic inequality*, Proc. Edinburgh Math. Soc. (2)**12**(1960/61), 139–147. MR**130334**, DOI 10.1017/S0013091500002777 - 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**, DOI 10.2140/pjm.1979.81.217 - 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 - B. A. Troesch,
*The shooting method applied to a cyclic inequality*, Math. Comp.**34**(1980), no. 149, 175–184. MR**551296**, DOI 10.1090/S0025-5718-1980-0551296-2 - B. A. Troesch,
*On Shapiro’s cyclic inequality for $N=13$*, Math. Comp.**45**(1985), no. 171, 199–207. MR**790653**, DOI 10.1090/S0025-5718-1985-0790653-0

*Notices Amer. Math. Soc.*, v. 29, 1982, pp. 130-133.

## Additional Information

- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp.
**53**(1989), 657-664 - MSC: Primary 26D15
- DOI: https://doi.org/10.1090/S0025-5718-1989-0983563-0
- MathSciNet review: 983563