Shapiro inequality

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

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

$\sum _{i=1}^n\frac{x_i}{x_{i+1}+x_{i+2}}\ge \frac{n}{2}$

where $x_{n+1}=x_1,x_{n+2}=x_2$.

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

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

The value of $\gamma$ was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound $\gamma$ is given by $\frac{1}{2}\psi (0)$, where the function $\psi$ is the convex hull of $f(x)=e^{-x}$ and $g(x)=\frac{2}{e^x+e^{\frac{x}{2}}}$. (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 $g$.)

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

Counter-examples for higher n

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

$x_{20}=(1+5ϵ,6ϵ,1+4ϵ,5ϵ,1+3ϵ,4ϵ,1+2ϵ,3ϵ,1+ϵ,2ϵ,1+2ϵ,ϵ,1+3ϵ,2ϵ,1+4ϵ,3ϵ,1+5ϵ,4ϵ,1+6ϵ,5ϵ)$ where $ϵ$ is close to 0.

Then the left-hand side is equal to $10-{ϵ}^2+O({ϵ}^3)$, thus lower than 10 when $ϵ$ 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

• 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 $n\le 12$, from which the result for all $n\le 12$ follows. They state $n=23$ as an open problem.

External links

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