Shapiro inequality

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

Statement of the inequality

Suppose [math]\displaystyle{ n }[/math] is a natural number and [math]\displaystyle{ x_1, x_2, \dots, x_n }[/math] are positive numbers and:

• [math]\displaystyle{ n }[/math] is even and less than or equal to [math]\displaystyle{ 12 }[/math], or
• [math]\displaystyle{ n }[/math] is odd and less than or equal to [math]\displaystyle{ 23 }[/math].

Then the Shapiro inequality states that

[math]\displaystyle{ \sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2} }[/math]

where [math]\displaystyle{ x_{n+1}=x_1, x_{n+2}=x_2 }[/math].

For greater values of [math]\displaystyle{ n }[/math] the inequality does not hold and the strict lower bound is [math]\displaystyle{ \gamma \frac{n}{2} }[/math] with [math]\displaystyle{ \gamma \approx 0.9891\dots }[/math].

The initial proofs of the inequality in the pivotal cases [math]\displaystyle{ n=12 }[/math] (Godunova and Levin, 1976) and [math]\displaystyle{ n=23 }[/math] (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for [math]\displaystyle{ n=12 }[/math].

The value of [math]\displaystyle{ \gamma }[/math] was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound [math]\displaystyle{ \gamma }[/math] is given by [math]\displaystyle{ \frac{1}{2} \psi(0) }[/math], where the function [math]\displaystyle{ \psi }[/math] is the convex hull of [math]\displaystyle{ f(x)=e^{-x} }[/math] and [math]\displaystyle{ g(x) = \frac{2}{e^x+e^{\frac{x}{2}}} }[/math]. (That is, the region above the graph of [math]\displaystyle{ \psi }[/math] is the convex hull of the union of the regions above the graphs of [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math].)

Interior local minima of the left-hand side are always [math]\displaystyle{ \ge\frac{n}2 }[/math] (Nowosad, 1968).

Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for [math]\displaystyle{ n=20 }[/math]:

[math]\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) }[/math] where [math]\displaystyle{ \epsilon }[/math] is close to 0.

Then the left-hand side is equal to [math]\displaystyle{ 10 - \epsilon^2 + O(\epsilon^3) }[/math], thus lower than 10 when [math]\displaystyle{ \epsilon }[/math] is small enough.

The following counter-example for [math]\displaystyle{ n=14 }[/math] is by Troesch (1985):

[math]\displaystyle{ x_{14} = (0, 42, 2, 42, 4, 41, 5, 39, 4, 38, 2, 38, 0, 40) }[/math] (Troesch, 1985)

References

• Fink, A.M. (1998). "Shapiro's inequality". in Gradimir V. Milovanović, G. V.. Recent progress in inequalities. Dedicated to Prof. Dragoslav S. Mitrinović. Mathematics and its Applications (Dordrecht). 430. Dordrecht: Kluwer Academic Publishers.. pp. 241–248. ISBN 0-7923-4845-1. 
• Bushell, P.J.; McLeod, J.B. (2002). "Shapiro's cyclic inequality for even n". J. Inequal. Appl. 7 (3): 331–348. ISSN 1029-242X. ftp://ftp.sam.math.ethz.ch/EMIS/journals/HOA/JIA/40a3.pdf.  They give an analytic proof of the formula for even [math]\displaystyle{ n\le12 }[/math], from which the result for all [math]\displaystyle{ n\le12 }[/math] follows. They state [math]\displaystyle{ n=23 }[/math] as an open problem.

External links

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