Advertisement

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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.MathSciNetGoogle Scholar
  5. 5.
    D. E. Daykin Inequalities for functions of a cyclic nature, J. London Math. Soc. (2) 3 (1971), 453–462.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    P. H. Diananda Extensions of an inequality of H. S. Shapiro, Amer. Math. Monthly 66 (1959), 489–491.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    P. H. Diananda, A cyclic inequality and an extension of it, I, Proc. Edinburgh Math. Soc. (2) 113 (1962/63), 79–84.MathSciNetCrossRefGoogle Scholar
  8. 8.
    P. H. Diananda, A cyclic inequality and an extension of it, II, Proc. Edinburgh Math. Soc. (2) 13 (1962/63), 143–152.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    P. H. Diananda, Inequalities for some cyclic sums, J. London Math. Soc. 38 (1963), 60–62.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    P. H. Diananda, On a cyclic sum, Proc. Glasgow Math. Assoc. 6 (1963), 11–13.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    P. H. Diananda, Inequalities for some cyclic sums, Math. Medley 5 (1977), 171–177.MathSciNetzbMATHGoogle Scholar
  12. 12.
    D. Ž. Djokovic Sur une inégalité, Proc. Glasgow Math. Assoc. 6 (1963), 1–10.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    V. G. Drinfel’d, A cyclic inequality, Mat. Zametki 9 (1971), 113–118 (Russian) [English transi. Math Notes 9 (1971), 68-71].MathSciNetGoogle Scholar
  14. 14.
    C. V. Durell, Query, Math. Gaz. 40 (1956), 266.CrossRefGoogle Scholar
  15. 15.
    A. M. Fink, Letter to the editor, Math. Gaz. 79 (1995), 125.CrossRefGoogle Scholar
  16. 16.
    E. S. Freidkin and S. A. Freidkin On a problèm by Shapiro, Elem. Math. 45 (1990), 137–139.MathSciNetzbMATHGoogle Scholar
  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].MathSciNetzbMATHGoogle Scholar
  18. 18.
    M. Herschern and J. E. L. Peck Problem 4603, Amer. Math. Monthly 67 (1960), 87–88.MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. A. Malcolm A note on a conjecture of L. J. Mordell, Math. Comp. 25 (1971), 375–377.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.zbMATHGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    R. A. Rankin An inequality, Math. Gaz. 42 (1958), 39–40.CrossRefGoogle Scholar
  26. 26.
    R. A. Rankin, A cyclic inequality, Proc. Edinburgh Math. Soc. (2) 12 (1960/61), 139–147.MathSciNetCrossRefGoogle Scholar
  27. 27.
    R. E. Scraton An unexpected minimum value, Math. Gaz. 78 (1994), 60–62.CrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    H. S. Shapiro, Problem 4603, Amer. Math. Monthly 61 (1954), 571.MathSciNetCrossRefGoogle Scholar
  31. 31.
    H. S. Shapiro, Problem 4603, Amer. Math. Monthly 63 (1956), 191–192.MathSciNetCrossRefGoogle Scholar
  32. 32.
    H. S. Shapiro, Problem 4603, Amer. Math. Monthly 97 (1990), 937.CrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  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.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    B. A. Troesch, On Shapiro’s cyclic inequality for N = 13, Math. Comp. 45 (1985), 199–207.MathSciNetzbMATHGoogle Scholar
  39. 39.
    B. A. Troesch, Full solution of Shapiro’s cyclic inequality, Notices Amer. Math. Soc. 39 (1985), no. 4, 318.MathSciNetGoogle Scholar
  40. 40.
    B. A. Troesch, The validity of Shapiro’s cyclic inequality, Math. Comp. 53 (1989), 657–664.MathSciNetzbMATHGoogle Scholar
  41. 41.
    A. Zulauf Note on a conjecture of L. J. Mordell, Abh. Math. Sem. Univ. Hamburg 22 (1958), 240–241.MathSciNetzbMATHGoogle Scholar
  42. 42.
    A. Zulauf On a conjecture of L. J. Mordell, II, Math. Gaz. 43 (1959), 182–184.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    A. Zulauf Note on an inequality, Math. Gaz. 46 (1962), 41–42.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • A. M. Fink
    • 1
  1. 1.Iowa State UniversityAmesUSA

Personalised recommendations