Shapiro inequality

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

Statement of the inequality

Suppose ${\displaystyle n}$ is a natural number and ${\displaystyle x_{1},x_{2},\dots ,x_{n}}$ are positive numbers and:

• ${\displaystyle n}$ is even and less than or equal to ${\displaystyle 12}$, or
• ${\displaystyle n}$ is odd and less than or equal to ${\displaystyle 23}$.

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 ${\displaystyle x_{n+1}=x_{1}}$ and ${\displaystyle x_{n+2}=x_{2}}$.

For greater values of ${\displaystyle n}$ the inequality does not hold, and the strict lower bound is ${\displaystyle \gamma {\frac {n}{2}}}$ with ${\displaystyle \gamma \approx 0.9891\dots }$(sequence A245330 in the OEIS ).

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

The value of ${\displaystyle \gamma }$ was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound ${\displaystyle \gamma }$ is given by ${\displaystyle \psi (0)}$, where the function ${\displaystyle \psi }$ is the convex hull of ${\displaystyle f(x)=e^{-x}}$ and ${\displaystyle g(x)={\frac {2}{e^{x}+e^{\frac {x}{2}}}}}$. (That is, the region above the graph of ${\displaystyle \psi }$ is the convex hull of the union of the regions above the graphs of ${\displaystyle f}$ and ${\displaystyle g}$.) ^{[5] [6]}

Interior local minima of the left-hand side are always ${\displaystyle \geq {\frac {n}{2}}}$. ^[7]

Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for ${\displaystyle n=20}$: ^[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 ${\displaystyle n=14}$ 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