Shapiro inequality

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

Statement of the inequality

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

• n is even and less than or equal to 12, or
• n is odd and less than or equal to 23.

Then the Shapiro inequality states that

${\displaystyle \sum _{i=1}^{n}{\frac {x_{i}}{x_{i+1}+x_{i+2}}}\geq {\frac {n}{2}},}$

where x_n+1 = x₁ and x_n+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 γ ⁠n/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) = e^−x and g(x) = 2 / (e^x + e^x/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]

${\displaystyle x_{20}=(1+5\epsilon ,\ 6\epsilon ,\ 1+4\epsilon ,\ 5\epsilon ,\ 1+3\epsilon ,\ 4\epsilon ,\ 1+2\epsilon ,\ 3\epsilon ,\ 1+\epsilon ,\ 2\epsilon ,\ 1+2\epsilon ,\ \epsilon ,\ 1+3\epsilon ,\ 2\epsilon ,\ 1+4\epsilon ,\ 3\epsilon ,\ 1+5\epsilon ,\ 4\epsilon ,\ 1+6\epsilon ,\ 5\epsilon ),}$

where ${\displaystyle \epsilon }$ is close to 0. Then the left-hand side is equal to ${\displaystyle 10-\epsilon ^{2}+O(\epsilon ^{3})}$, thus lower than 10 when ${\displaystyle \epsilon }$ is small enough.

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

${\displaystyle x_{14}=(0,42,2,42,4,41,5,39,4,38,2,38,0,40)}$ (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.

• Fink, A.M. (1998). "Shapiro's inequality". In Gradimir V. Milovanović, G. V. (ed.). Recent progress in inequalities. Dedicated to Prof. Dragoslav S. Mitrinović. Mathematics and its Applications (Dordrecht). Vol. 430. Dordrecht: Kluwer Academic Publishers. pp. 241–248. ISBN 0-7923-4845-1. Zbl 0895.26001.

External links

• Usenet discussion in 1999 (Dave Rusin's notes)
• PlanetMath