Shapiro inequality

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In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954. [1]

Contents

Statement of the inequality

Suppose n is a natural number and x1, x2, , xn are positive numbers and:

Then the Shapiro inequality states that

where xn+1 = x1 and xn+2 = x2. The special case with n = 3 is Nesbitt's inequality.

For greater values of n the inequality does not hold, and the strict lower bound is γn/2 with γ 0.9891(sequence A245330 in the OEIS ).

The initial proofs of the inequality in the pivotal cases n = 12 [2] and n = 23 [3] rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for n = 12. [4]

The value of γ was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound γ is given by ψ(0), where the function ψ is the convex hull of f(x) = ex and g(x) = 2 / (ex + ex/2). (That is, the region above the graph of ψ is the convex hull of the union of the regions above the graphs of f and g.) [5] [6]

Interior local minima of the left-hand side are always n / 2. [7]

Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for n = 20: [8]

where is close to 0. Then the left-hand side is equal to , thus lower than 10 when is small enough.

The following counter-example for n = 14 is by Troesch (1985):

(Troesch, 1985)

References

  1. Shapiro, H. S.; Bellman, R.; Newman, D. J.; Weissblum, W. E.; Smith, H. R.; Coxeter, H. S. M. (1954). "Problems for Solution: 4603-4607" . The American Mathematical Monthly. 61 (8): 571. doi:10.2307/2307617. JSTOR   2307617 . Retrieved 2021-09-23.
  2. Godunova, E. K.; Levin, V. I. (1976-06-01). "A cyclic sum with 12 terms" . Mathematical Notes of the Academy of Sciences of the USSR. 19 (6): 510–517. doi:10.1007/BF01149930. ISSN   1573-8876.
  3. Troesch, B. A. (1989). "The Validity of Shapiro's Cyclic Inequality" . Mathematics of Computation. 53 (188): 657–664. doi:10.2307/2008728. ISSN   0025-5718. JSTOR   2008728.
  4. Bushell, P. J.; McLeod, J. B. (2002). "Shapiro's cyclic inequality for even n". Journal of Inequalities and Applications. 7 (3): 331–348. doi: 10.1155/S1025583402000164 . ISSN   1029-242X.
  5. Drinfel'd, V. G. (1971-02-01). "A cyclic inequality" . Mathematical Notes of the Academy of Sciences of the USSR. 9 (2): 68–71. doi:10.1007/BF01316982. ISSN   1573-8876. S2CID   121786805.
  6. Weisstein, Eric W. "Shapiro's Cyclic Sum Constant". MathWorld .
  7. Nowosad, Pedro (September 1968). "Isoperimetric eigenvalue problems in algebras" . Communications on Pure and Applied Mathematics. 21 (5): 401–465. doi:10.1002/cpa.3160210502. ISSN   0010-3640.
  8. Lighthill, M. J. (1956). "An Invalid Inequality" . American Mathematical Monthly. 63 (3): 191–192. doi:10.1080/00029890.1956.11988785.