Shapiro inequality

In mathematics, the Shapiro inequality is an inequality proposed by H. 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 $x_{n+1}=x_{1},x_{n+2}=x_{2}$.

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, who won a Fields Medal in 1990. Specifically, Drinfeld showed that the strict lower bound $\gamma$ is given by ${\frac {1}{2}}\psi (0)$, where $\psi$ is the function 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 mimima of the left-hand side are always$\geq {\frac {n}{2}}$ (Nowosad, 1968).

Counter-examples for higher $n$

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

$x_{20}=(1+5\epsilon ,\ 6\epsilon ,\ 1+4\epsilon ,\ 5\epsilon ,\ 1+3\epsilon ,\ 4\epsilon ,\ 1+2\epsilon ,\ 3\epsilon ,\ 1+\epsilon ,\ 2\epsilon ,\ 1+2\epsilon ,\ \epsilon ,\ 1+3\epsilon ,\ 2\epsilon ,\ 1+4\epsilon ,\ 3\epsilon ,\ 1+5\epsilon ,\ 4\epsilon ,\ 1+6\epsilon ,\ 5\epsilon )$ 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.
• Bushell, P.J.; McLeod, J.B. (2002). "Shapiro's cyclic inequality for even n" (PDF). J. Inequal. Appl. 7 (3): 331–348. ISSN 1029-242X. Zbl 1018.26010. They give an analytic proof of the formula for even $n\leq 12$, from which the result for all $n\leq 12$ follows. They state $n=23$ as an open problem.

External links

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