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

:$\sum_^ \frac \geq \frac$

where $x_=x_1, x_=x_2$.

For greater values of $n$ the inequality does not hold and the strict lower bound is $\gamma \frac$ 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 $\frac \psi(0)$, where the function $\psi$ is the convex hull of $f(x)=e^$ and $g(x) = \frac$. (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$\ge\frac2$ (Nowosad, 1968).

Counter-examples for higher ''n''

The first counter-example was found by Lighthill in 1956, for $n=20$:
:$x_ = (1+5\epsilon,\ 6\epsilon,\ 1+4\epsilon,\ 5\epsilon,\ 1+3\epsilon,\ 4\epsilon,\ 1+2\epsilon,\ 3\epsilon,\ 1+\epsilon,\ 2\epsilon,\ 1+2\epsilon,\ \epsilon,\ 1+3\epsilon,\ 2\epsilon,\ 1+4\epsilon,\ 3\epsilon,\ 1+5\epsilon,\ 4\epsilon,\ 1+6\epsilon,\ 5\epsilon)$ 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):
:$x_ = (0, 42, 2, 42, 4, 41, 5, 39, 4, 38, 2, 38, 0, 40)$ (Troesch, 1985)

References

*
* {{cite journal , zbl=1018.26010 , last1=Bushell , first1=P.J. , last2=McLeod , first2=J.B. , title=Shapiro's cyclic inequality for even n , journal=J. Inequal. Appl. , volume=7 , number=3 , pages=331–348 , year=2002 , issn=1029-242X , url=ftp://ftp.sam.math.ethz.ch/EMIS/journals/HOA/JIA/40a3.pdf They give an analytic proof of the formula for even $n\le12$, from which the result for all $n\le12$ follows. They state $n=23$ as an open problem.

External links

Usenet discussion in 1999
(Dave Rusin's notes)



Inequalities