Shapiro inequality explained

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,} ... , x _{n} are positive numbers and:n is even and less than or equal to 12 , orn is odd and less than or equal to 23 .

Then the Shapiro inequality states that

\sum ^{n} _{i =1} (x _{i})/(x _{i+1} + x _{i+2}) \geq (n)/(2)

where

x _{n+1} = x _{1} and x _{n+2} = x _{2} .

For greater values of

n the inequality does not hold, and the strict lower bound is \gamma (n)/(2) with \gamma ≈ 0 . 9891... .

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

\gamma was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound \gamma is given by (1)/(2) \psi( 0 ) , where the function \psi 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 \psi is the convex hull of the union of the regions above the graphs of f and g .)^[5]

Interior local minima of the left-hand side are always

\ge (n)/(2) .^[6]

Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for

n =20 :^[7] 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

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

External links

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

Notes and 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 . 10.2307/2307617 . 2307617 . 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 . en . 19 . 6 . 510–517 . 10.1007/BF01149930 . 1573-8876.
3. Troesch . B. A. . 1989 . The Validity of Shapiro's Cyclic Inequality . Mathematics of Computation . 53 . 188 . 657–664 . 10.2307/2008728 . 2008728 . 0025-5718.
4. Bushell . P. J. . McLeod . J. B. . 2002 . Shapiro's cyclic inequality for even n . Journal of Inequalities and Applications . English . 7 . 3 . 331–348 . 10.1155/S1025583402000164 . free . 1029-242X.
5. Drinfel'd. V. G.. 1971-02-01. A cyclic inequality. Mathematical Notes of the Academy of Sciences of the USSR. en. 9. 2. 68–71. 10.1007/BF01316982. 121786805. 1573-8876.
6. Nowosad . Pedro . September 1968 . Isoperimetric eigenvalue problems in algebras . Communications on Pure and Applied Mathematics . en . 21 . 5 . 401–465 . 10.1002/cpa.3160210502 . 0010-3640.
7. Lighthill . M. J. . 1956 . An Invalid Inequality . American Mathematical Monthly . 63 . 3 . 191–192. 10.1080/00029890.1956.11988785 .