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 $x 1, x 2, \dots, x n$ 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

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

where $x n +1 = x 1$ and $x n +2 = x 2$ . 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 $γ .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠n/2⁠$ with $γ \approx 0.9891\dots$ (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 $\geq n / 2$ .^[7]

Counter-examples for higher n

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

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

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

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

(Troesch, 1985)

References

^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ Weisstein, Eric W. "Shapiro's Cyclic Sum Constant". MathWorld.
^ 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.
^ 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

[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.

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