# Shapiro’s Inequality

• A. M. Fink
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 430)

## Abstract

Shapiro proposed an inequality in the 1954 American Mathematical Monthly, which now goes by his name. The inequality is now settled but work on the subject continues. The discussion of the history of this inequality does not always give a clear picture of the chronology and the results. In addition, the work still to be done is not always made clear. Here, we try to separate the various aspects of the problem: give the chronology and priorities; give a hint on the role of computer proofs; give the shortest route to the results; indicate the methods used in the shortest route; conjecture about the continued interest in the problem; and indicate where further work can be done.

## Key words and phrases

Cyclic inequality History

## References

1. 1.
B. Bajšanski, A remark concerning the lower bound of x 1/(x 2 + x 3) + x 2/(x 3 + x 4) +. + xn/(x 1 + x 2), Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Math. Fiz. No. 70-76 (1962), 19–20.Google Scholar
2. 2.
P. J. Bushell, Shapiro’s cyclic sum, manuscript.Google Scholar
3. 3.
P. J. Bushell, Analytic proofs of Shapiro’s cyclic inequality for even n, manuscript.Google Scholar
4. 4.
P. J. Bushell and A. M. Craven, On Shapiro’s cyclic inequality, Proc. Roy. Soc. Edinburgh 75A26 (1975/76), 333–338.
5. 5.
D. E. Daykin Inequalities for functions of a cyclic nature, J. London Math. Soc. (2) 3 (1971), 453–462.
6. 6.
P. H. Diananda Extensions of an inequality of H. S. Shapiro, Amer. Math. Monthly 66 (1959), 489–491.
7. 7.
P. H. Diananda, A cyclic inequality and an extension of it, I, Proc. Edinburgh Math. Soc. (2) 113 (1962/63), 79–84.
8. 8.
P. H. Diananda, A cyclic inequality and an extension of it, II, Proc. Edinburgh Math. Soc. (2) 13 (1962/63), 143–152.
9. 9.
P. H. Diananda, Inequalities for some cyclic sums, J. London Math. Soc. 38 (1963), 60–62.
10. 10.
P. H. Diananda, On a cyclic sum, Proc. Glasgow Math. Assoc. 6 (1963), 11–13.
11. 11.
P. H. Diananda, Inequalities for some cyclic sums, Math. Medley 5 (1977), 171–177.
12. 12.
D. Ž. Djokovic Sur une inégalité, Proc. Glasgow Math. Assoc. 6 (1963), 1–10.
13. 13.
V. G. Drinfel’d, A cyclic inequality, Mat. Zametki 9 (1971), 113–118 (Russian) [English transi. Math Notes 9 (1971), 68-71].
14. 14.
C. V. Durell, Query, Math. Gaz. 40 (1956), 266.
15. 15.
A. M. Fink, Letter to the editor, Math. Gaz. 79 (1995), 125.
16. 16.
E. S. Freidkin and S. A. Freidkin On a problèm by Shapiro, Elem. Math. 45 (1990), 137–139.
17. 17.
E. K. Godunova and V. I. Levin, A cyclic sum with 12 terms, Mat. Zametki 19 (1976), 873–885 (Russian) [English transi. Math Notes 19 (1976), 510-517].
18. 18.
M. Herschern and J. E. L. Peck Problem 4603, Amer. Math. Monthly 67 (1960), 87–88.
19. 19.
M. A. Malcolm A note on a conjecture of L. J. Mordell, Math. Comp. 25 (1971), 375–377.
20. 20.
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.
21. 21.
L. J. Mordell On the inequality $$\sum\nolimits_{r = 1}^n {{x_r}} /({x_{r + 1}} + {x_{r + 2}}) \geqq n/2$$ and some others, Abh. Math. Sem. Univ. Hamburg 22 (1958), 229–240.
22. 22.
L. J. Mordell Note on the inequality $$\sum\nolimits_{k = 1}^n {{x_k}} /({x_{k + 1}} + {x_{k + 2}}) \geqslant n/2$$, J. London Math. Soc. 37 (1962), 176–178.
23. 23.
A. M. Nesbitt Problem 15114, Educational Times (2) 3 (1903), 37–38.Google Scholar
24. 24.
P. Nowosad Isoperimetric eigenvalue problems in algebras, Comm. Pure Appl. Math. 21 (1968), 401–465.
25. 25.
R. A. Rankin An inequality, Math. Gaz. 42 (1958), 39–40.
26. 26.
R. A. Rankin, A cyclic inequality, Proc. Edinburgh Math. Soc. (2) 12 (1960/61), 139–147.
27. 27.
R. E. Scraton An unexpected minimum value, Math. Gaz. 78 (1994), 60–62.
28. 28.
J. L. Searcy and B. A. Troesch, The cyclic inequality, Notices Amer. Math. Soc. 23 (1976), 604–605.Google Scholar
29. 29.
J. L. Searcy and B. A. Troesch, A cyclic inequality and related eigenvalue problèm, Pacific J. Math. 81 (1979), 217–226.
30. 30.
H. S. Shapiro, Problem 4603, Amer. Math. Monthly 61 (1954), 571.
31. 31.
H. S. Shapiro, Problem 4603, Amer. Math. Monthly 63 (1956), 191–192.
32. 32.
H. S. Shapiro, Problem 4603, Amer. Math. Monthly 97 (1990), 937.
33. 33.
J. Stuart, On Kristiansen’s Proof of Shapiro’s Inequality for n = 12, Diss. Univ. of Reading, 1974 (not published).Google Scholar
34. 34.
D. G. S. Thomas On the definiteness of certain quadratic forms arising in a conjecture of L. J. Mordell, Amer. Math. Monthly 68 (1961), 472–473.
35. 35.
B. A. Troesch, The cyclic inequality for a large number of terms, Notices Amer. Math. Soc. 25 (1978), no. 6, A–627.Google Scholar
36. 36.
B. A. Troesch, Shapiro’s cyclic inequality with eleven terms, Notices Amer. Math. Soc. 26 (1979), no. 7, A–646.Google Scholar
37. 37.
B. A. Troesch, The shooting method applied to a cyclic inequality, Math. Comp. 34 (1980), 175–184.
38. 38.
B. A. Troesch, On Shapiro’s cyclic inequality for N = 13, Math. Comp. 45 (1985), 199–207.
39. 39.
B. A. Troesch, Full solution of Shapiro’s cyclic inequality, Notices Amer. Math. Soc. 39 (1985), no. 4, 318.
40. 40.
B. A. Troesch, The validity of Shapiro’s cyclic inequality, Math. Comp. 53 (1989), 657–664.
41. 41.
A. Zulauf Note on a conjecture of L. J. Mordell, Abh. Math. Sem. Univ. Hamburg 22 (1958), 240–241.
42. 42.
A. Zulauf On a conjecture of L. J. Mordell, II, Math. Gaz. 43 (1959), 182–184.
43. 43.
A. Zulauf Note on an inequality, Math. Gaz. 46 (1962), 41–42.