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

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

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 $\gamma {\frac {n}{2}}$ with $\gamma \approx 0.9891\dots$.

The initial proofs of the inequality in the pivotal cases $n=12$ and $n=23$ 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 $\geq {\frac {n}{2}}$.

Counter-examples for higher n

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

$x_{20}=(1+5\epsilon ,\ 6\epsilon ,\ 1+4\epsilon ,\ 5\epsilon ,\ 1+3\epsilon ,\ 4\epsilon ,\ 1+2\epsilon ,\ 3\epsilon ,\ 1+\epsilon ,\ 2\epsilon ,\ 1+2\epsilon ,\ \epsilon ,\ 1+3\epsilon ,\ 2\epsilon ,\ 1+4\epsilon ,\ 3\epsilon ,\ 1+5\epsilon ,\ 4\epsilon ,\ 1+6\epsilon ,\ 5\epsilon ),$

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

• 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