The validity of Shapiro's cyclic inequality

Author:
B. A. Troesch

Journal:
Math. Comp. **53** (1989), 657-664

MSC:
Primary 26D15

DOI:
https://doi.org/10.1090/S0025-5718-1989-0983563-0

MathSciNet review:
983563

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A cyclic sum is formed with *N* components of a vector x, where in the sum , , 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 15 and 23, this has apparently never been proved. Here we will confirm that the inequality indeed holds for all odd . This settles the question for all *N*.

**[1]**P. H. Diananda, "On a cyclic sum,"*Proc. Glasgow Math. Assoc.*, v. 6, 1963, pp. 11-13. MR**0150084 (27:87)****[2]**P. H. Diananda, "A cyclic inequality and an extension of it. II,"*Proc. Edinburgh Math. Soc.*(2), v. 13, 1962, pp. 143-152. MR**0148831 (26:6335)****[3]**D. Ž. Djoković, "Sur une inégalité,"*Proc. Glasgow Math. Assoc.*, v. 6, 1963, pp. 1-10. MR**0150083 (27:86)****[4]**V. G. Drinfeld, "A cyclic inequality,"*Math. Notes*, v. 9, 1971, pp. 68-71. MR**0280660 (43:6379)****[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. Lagarias, "The van der Waerden conjecture: Two Soviet solutions,"*Notices Amer. Math. Soc.*, v. 29, 1982, pp. 130-133.**[8]**D. S. Mitrinović,*Analytic Inequalities*, Springer-Verlag, New York, 1970. MR**0274686 (43:448)****[9]**P. Novosad, "Isoperimetric eigenvalue problems in algebra,"*Comm. Pure Appl. Math.*, v. 21, 1968, pp. 401-465. MR**0238087 (38:6363)****[10]**R. A. Rankin, "A cyclic inequality,"*Proc. Edinburgh Math. Soc.*(2), v. 12, 1961, pp. 139-147. MR**0130334 (24:A198)****[11]**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)****[12]**H. S. Shapiro, "Problem 4603,"*Amer. Math. Monthly*, v. 61, 1954, p. 571. MR**1528827****[13]**B. A. Troesch, "The shooting method applied to a cyclic inequality,"*Math. Comp.*, v. 34, 1980, pp. 175-184. MR**551296 (81b:10022)****[14]**B. A. Troesch, "On Shapiro's cyclic inequality for ,"*Math. Comp.*, v. 45, 1985, pp. 199-207. MR**790653 (87h:26031)**

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

Retrieve articles in all journals with MSC: 26D15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0983563-0

Keywords:
Cyclic inequality,
cyclic sum,
minimization

Article copyright:
© Copyright 1989
American Mathematical Society