On Shapiro's cyclic inequality for

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A cyclic sum is formed with the *N* components of a vector **x**, where , , and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors **x** for which if and even, and if . It has been proved that the inequality holds for . 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 .

**[1]**P. J. Bushell and A. H. Craven,*On Shapiro’s cyclic inequality*, Proc. Roy. Soc. Edinburgh Sect. A**75**(1975/76), no. 4, 333–338. MR**0480320**, https://doi.org/10.1017/S0308210500013366**[2]**D. E. Daykin,*Inequalities for functions of a cyclic nature*, J. London Math. Soc. (2)**3**(1971), 453–462. MR**0284394**, https://doi.org/10.1112/jlms/s2-3.3.453**[3]**P. H. Diananda,*On a cyclic sum*, Proc. Glasgow Math. Assoc.**6**(1963), 11–13 (1963). MR**0150084****[4]**D. Ž. Djoković,*Sur une inégalité*, Proc. Glasgow Math. Assoc.**6**(1963), 1–10 (1963) (French). MR**0150083****[5]**J. Lambek and L. Moser,*Rational analogues of the logarithm function*, Math. Gaz.**40**(1956), 5–7. MR**0075977**, https://doi.org/10.2307/3610258**[6]**E. K. Godunova and V. I. Levin,*A cyclic sum with twelve terms*, Mat. Zametki**19**(1976), no. 6, 873–885 (Russian). MR**0424578****[7]**J. C. Lagartas, "The van der Waerden conjecture: Two Soviet solutions,"*Notices Amer. Math. Soc.*, v. 29, 1982, pp. 130-133.**[8]**Michael A. Malcolm,*A note on a conjecture of L. J. Mordell*, Math. Comp.**25**(1971), 375–377. MR**0284395**, https://doi.org/10.1090/S0025-5718-1971-0284395-8**[9]**D. S. Mitrinović,*Analytic inequalities*, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR**0274686****[10]**L. J. Mordell,*On the inequality ∑ⁿᵣ₌₁𝑥ᵣ/(𝑥ᵣ₊₁+𝑥ᵣ₊₂)≧𝑛/2 and some others*, Abh. Math. Sem. Univ. Hamburg**22**(1958), 229–241. MR**0096915**, https://doi.org/10.1007/BF02941955**[11]**Pedro Nowosad,*Isoperimetric eigenvalue problems in algebras*, Comm. Pure Appl. Math.**21**(1968), 401–465. MR**0238087**, https://doi.org/10.1002/cpa.3160210502**[12]**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****[13]**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**, https://doi.org/10.2307/2307617**[14]**A. Zulauf,*On a conjecture of L. J. Mordell*, Abh. Math. Sem. Univ. Hamburg**22**(1958), 240–241. MR**0124261**, https://doi.org/10.2307/3610955

Retrieve articles in *Mathematics of Computation*
with MSC:
26D15,
05A20

Retrieve articles in all journals with MSC: 26D15, 05A20

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